mm-3579 === Subject: Re: A question on algebraic circle fitting I was able to extract the answers to my questions from them: >> Given a finite number of points in the plane, >> (x1,y1), (x2,y2), ..., (xn,yn), >> the algebraic way to fit a circle to these points >> is to minimize the sum >> sum( (xi^2 + yi^2 + 2 D xi + 2 E yi + F)^2 , i = 1..n ) >>over the variables D,E,F. >>Is it correct that the resulting minimizers >> will satisfy the condition >> D^2 + E^2 >= F ? This is correct: Instead of minimizing over the variables D, E, and F, one can as well first minimize over D and E, then introduce a new variable q := D^2 + E^2 - F, and in terms of this variable, the minimization problem for F turns into minimizing the sum sum( ( (xi+D)^2 + (yi+E)^2 - q )^2 , i = 1..n ) over q. The solution to this problem, q = sum( (xi+D)^2 + (yi+E)^2 , i = 1..n ) / n, is obviously non-negative. This also shows that this method of algebraic circle fittiing and the method of geometric circle fitting- minimizing the sum of squares of distances from the circle to the given data points - will never yield the same result, except when the data points are concircular. The reason for this is that, as seen above, algebraic circle fitting yields an optimal circle of a radius equal to the geometric mean of the distances of the points (xi,yi) to the midpoint (-D,-E) of the optimal circle, while geometric circle fitting yields a circle of radius equal to the arithmetic mean of these distances. - Well, not exactly the same distances, because usually the optimum circle midpoints for the geometric method are different from those for the algebraic one. In particular, it is no contradiction that often in examples the optimal radii for the geometric method are larger than for the algebraic one, although the arithmetic mean of n nonpositive numbers is never larger than their geometric mean. >> Under what conditions will minimizers D,E,F exist? - >> A necessary condition is that not all points (xi,yi) lie on >> a common line in case n >= 3. Is this condition also sufficient? Yes, it is. - The only possibility for minimizing circles not to exist, besides all data points being equal, is that the covariance matrix of the pairs (xi,yi) becomes singular, which only happens iff the points are collinear. Once again, thank you Thomas === Subject: Re: Proyective Module... easy? An approach that generalizes to all the number fields and all the > ideals of their rings of integers (or Dedekind domains). Let P' > be the inverse of P in the class group. Then PP'=R, so we can > write 1 as a finite sum Oops. I meant that let P' be the fractional ideal that is the inverse of P in the group of fractional ideals of R. Of course, if P and Q are two ideals in the same class, they are isomorphic as modules (P isomorphic to aP etc.), so one can pass over to the class group, but I should have explained this bit. Good luck! Jyrki === Subject: Minimization problem I have a minimization problem that I want to solve: 1) Define X as an N x M matrix (with positive values if it matters) 2) Define w as an Mx1 weight vector (with positive values if it matters). We can now construct a weighted mean of X, weighted mean = Xw 3) Find the Mx1 weight vector v that minimizes the total squared error, toterr = (w-v)' X' X (w-v) given than v can only have L (L=1,...,M) non-zero entries. In other words, I want to find a subvector v of w that captures most of the variance, given that a lot of the entries in v are zero. Any pointers or ideas how I can solve this? Eric === Subject: Re: Minimization problem I have a minimization problem that I want to solve: 1) Define X as an N x M matrix (with positive values if it matters) > 2) Define w as an Mx1 weight vector (with positive values if it > matters). > We can now construct a weighted mean of X, weighted mean = Xw > 3) Find the Mx1 weight vector v that minimizes the total squared error, > toterr = (w-v)' X' X (w-v) given than v can only have L (L=1,...,M) non-zero entries. In other words, I want to find a subvector v of w that captures most of > the variance, given that a lot of the entries in v are zero. Any pointers or ideas how I can solve this? Eric It's perhaps easier if I tell the background. A stock index is composed of M stocks with different fixed weights. I want to find the L stocks and corresponding weights that minimize the difference between my portfolio (the L stocks) and the stock index. Eric === Subject: Moebius Band is not homeomorphic with a Torus Moebius Band is not homeomorphic with a Torus or even orientable like it. Mathematica formula and images of the unifying set of tori in toroidal co-ordinate system I have indicated in: http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg I made a square section steel ring of m = 2 configuration by cut/twist/weld and feel that the surface morphology of all these surfaces has a unifying character by its fiber bundle.Of course the standard Moebius Band is made by cut/twist/paste of a rectangular sheet having two sides, small thickness of the sheet is conveniently overlooked to result in one side only! There are actually four sides in all and two tracks. By varying m or Aspect Ratio of a rectangular section of a tube (or solid section) of torus one can morph/animate images among all tori including the Moebius Band. I am not comfortable with the Moebius Band singled out topologically either in terms of orientability or homeomorphability. What exactly am I missing? Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus The following experiment was performed on an automobile wheel rubber tube to consider role of both sides of the surface necessary to fully describe the surface. The tube was filled with thermosetting adhesive. Air is gradually evacuated from the inside of the tube, keeping inclination at four quarter points to ground plane at 0, + 45, 90, + 45 degrees at quarter points with local stretching as needed. The tube is heated and inside surfaces allowed to stick together so that there is no more any inside due to a monolithic rubber sheet formation of double thickness.The sketches show how opposite points are brought together (or just joined if you like) to make an MB out of a Torus. For MB visibility only a half of the Torus is shown. http://img79.imageshack.us/img79/8162/mbtorussl6.jpg What results by this process is a Moebius strip that is supposedly non-orientable when converted from the oriented Torus surface. The only new requirement is that 720 degrees of polar rotation is needed instead of 360 degrees to come back to the same point with same direction/sense of normal vector without crossing the band edges. Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus The following experiment was performed on an automobile wheel rubber tube to consider role of both sides of the surface necessary to fully describe the surface. The tube was filled with thermosetting adhesive. Air is gradually evacuated from the inside of the tube, keeping inclination at four quarter points to ground plane at 0, + 45, 90, + 45 degrees at quarter points with local stretching as needed. The tube is heated and inside surfaces allowed to stick together so that there is no more any inside due to a monolithic rubber sheet formation of double thickness.The sketches show how opposite points are brought together (or just joined if you like) to make an MB out of a Torus. For MB visibility only a half of the Torus is shown. http://img79.imageshack.us/img79/8162/mbtorussl6.jpg What results by this process is a Moebius strip that is supposedly non-orientable when converted from the oriented Torus surface. The only new requirement is that 720 degrees of polar rotation is needed instead of 360 degrees to come back to the same point with same direction/sense of normal vector without crossing the band edges. Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus > Moebius Band is not homeomorphic with a Torus or even orientable like > it. However you define a Mobius strip (with or without a boundary) its fundamental group is cyclic and that of the torus isn't. For each of these surfaces it's easy to write down explicitly the universal cover and this gets the fundamental groups immediately. > I made a square section steel ring of m = 2 configuration by > cut/twist/weld and feel that the surface morphology of all these > surfaces has a unifying character by its fiber bundle. Neither your steel constructions nor your feelings for surface morphology have any mathematical force. > What exactly am I missing? Possibly the desire or the ability to think mathematically? Victor Meldrew === Subject: Re: Moebius Band is not homeomorphic with a Torus > Moebius Band is not homeomorphic with a Torus or even orientable like > it. Mathematica formula and images of the unifying set of tori in toroidal > co-ordinate system I have indicated in: http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg I made a square section steel ring of m = 2 configuration by > cut/twist/weld and feel that the surface morphology of all these > surfaces has a unifying character by its fiber bundle.Of course the > standard Moebius Band is made by cut/twist/paste of a rectangular sheet > having two sides, small thickness of the sheet is conveniently > overlooked to result in one side only! There are actually four sides > in all and two tracks. > No, a model made from steel has thickness; a Moebius Band has no width, anymore than does a mathematical line. === Subject: Re: Moebius Band is not homeomorphic with a Torus Narasimham napisal(a): > Moebius Band is not homeomorphic with a Torus or even orientable like > it. Mathematica formula and images of the unifying set of tori in toroidal > co-ordinate system I have indicated in: http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg I made a square section steel ring of m = 2 configuration by > cut/twist/weld and feel that the surface morphology of all these > surfaces has a unifying character by its fiber bundle.Of course the > standard Moebius Band is made by cut/twist/paste of a rectangular sheet > having two sides, small thickness of the sheet is conveniently > overlooked to result in one side only! There are actually four sides > in all and two tracks. By varying m or Aspect Ratio of a rectangular section of a tube (or > solid section) of torus one can morph/animate images among all tori > including the Moebius Band. I am not comfortable with the Moebius Band singled out topologically > either in terms of orientability or homeomorphability. What exactly am I missing? Narasimham There are many ways to show that MB is not homeomorphic with T. For example: Torus is a surface, Moebius Band is not a surface. Hox === Subject: Re: Moebius Band is not homeomorphic with a Torus > Narasimham napisal(a): > Moebius Band is not homeomorphic with a Torus or even orientable like > it. > Mathematica formula and images of the unifying set of tori in toroidal > co-ordinate system I have indicated in: > http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg > I made a square section steel ring of m = 2 configuration by > cut/twist/weld and feel that the surface morphology of all these > surfaces has a unifying character by its fiber bundle.Of course the > standard Moebius Band is made by cut/twist/paste of a rectangular sheet > having two sides, small thickness of the sheet is conveniently > overlooked to result in one side only! There are actually four sides > in all and two tracks. > By varying m or Aspect Ratio of a rectangular section of a tube (or > solid section) of torus one can morph/animate images among all tori > including the Moebius Band. > I am not comfortable with the Moebius Band singled out topologically > either in terms of orientability or homeomorphability. > What exactly am I missing? > Narasimham > There are many ways to show that MB is not homeomorphic with T. For example: Torus is a surface, Moebius Band is not a surface. > I'd call it a surface. What's your definition of surface? One distinction is that a Moebius band has boundary, whereas a torus doesn't. > Hox === Subject: Re: Moebius Band is not homeomorphic with a Torus > Narasimham napisal(a): >> Moebius Band is not homeomorphic with a Torus or even orientable like >> it. >>Mathematica formula and images of the unifying set of tori in toroidal >> co-ordinate system I have indicated in: >>http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg >>I made a square section steel ring of m = 2 configuration by >> cut/twist/weld and feel that the surface morphology of all these >> surfaces has a unifying character by its fiber bundle.Of course the >> standard Moebius Band is made by cut/twist/paste of a rectangular sheet >> having two sides, small thickness of the sheet is conveniently >> overlooked to result in one side only! There are actually four sides >> in all and two tracks. >>By varying m or Aspect Ratio of a rectangular section of a tube (or >> solid section) of torus one can morph/animate images among all tori >> including the Moebius Band. >>I am not comfortable with the Moebius Band singled out topologically >> either in terms of orientability or homeomorphability. >>What exactly am I missing? >>Narasimham >> There are many ways to show that MB is not homeomorphic with T. >> For example: Torus is a surface, Moebius Band is not a surface. I'd call it a surface. What's your definition of surface? One distinction is that a Moebius band has boundary, whereas a torus > doesn't. > Hox > Not that I disagree with your argument, but can a Mobius Band have infinite width and therefore be free of boundaries? === Subject: Re: Moebius Band is not homeomorphic with a Torus > Narasimham napisal(a): >Moebius Band is not homeomorphic with a Torus or even orientable like >it. > Mathematica formula and images of the unifying set of tori in toroidal >co-ordinate system I have indicated in: > http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg > I made a square section steel ring of m = 2 configuration by >cut/twist/weld and feel that the surface morphology of all these >surfaces has a unifying character by its fiber bundle.Of course the >standard Moebius Band is made by cut/twist/paste of a rectangular sheet >having two sides, small thickness of the sheet is conveniently >overlooked to result in one side only! There are actually four sides >in all and two tracks. > By varying m or Aspect Ratio of a rectangular section of a tube (or >solid section) of torus one can morph/animate images among all tori >including the Moebius Band. > I am not comfortable with the Moebius Band singled out topologically >either in terms of orientability or homeomorphability. > What exactly am I missing? > Narasimham > > There are many ways to show that MB is not homeomorphic with T. > For example: Torus is a surface, Moebius Band is not a surface. >I'd call it a surface. What's your definition of surface? > Surface is a 2-dimensional manifold. In other words, for every point x of space Y, there must exist a neighborhood (an open set, with point x) in Y, homeomorphic with E^2 (Euclidian 2-dim space) > One distinction is that a Moebius band has boundary, whereas a torus > doesn't. > And so, for a point x from boundary of MB, there are no neighborhood of x in MB, that could satisfy above condition. > Hox === Subject: Re: Moebius Band is not homeomorphic with a Torus And so, for a point x from boundary of MB, there are > no neighborhood of x in MB, > that could satisfy above condition. Who said the points on the boundary are part of MB? > The *open* MB is a 2-manifold. It's not compact, > it's not orientable, so there are plenty of ways > of telling it's not homeomorphic (or diffeomorphic > if you prefer that) to the torus, but it is in this > sense a 2-manifold. > Jyrki > One distinction is that a Moebius band has boundary, whereas a torus > doesn't. I think that words has boundary means that it has boundary :-) and we are talking about *close* MB. Hox === Subject: Re: Translates in R^2 and lattices Another idea: Take the intersections of R with the unit squares of the lattice and translate them into the fundamental square P. Then some point x of P is covered by at least |A| translates, so R-x contains at least |A| integer points. Simeon > Some advice, please: If R is a region in R^2, with finite area A, > how do we show that we can translate R so that > R will contain at least : > _ _ > | A | lattice points? (i.e, the smallest integer > greater or equal to A). > Thanx === Subject: Re: almost surely convergence and infitely often... > Hi all, >> Could anybody show how to rigorously show if the following is true: >> If Ln / n -> 1 almost surely, as n -> infinity, where Ln is a > sequence of random variable, > then P(Ln >=n i.o.) = 1... >> where i.o. means the event occurs infinitely often. >> Is the above true? And how to show it? > It looks definitely not true, since you can have situations where Ln >> is never be greater than or equal to n and still have Ln / n -> 1 >> almost surely. >> David Jones So what's your opinion about P(Ln >=n i.o.) ? Should it be P(Ln >=n i.o.)=0? I knew it should be either 0 or 1. Based on your comment, I am now confused. > I have to admit that I don't know. As Robert Israel said in his reply, if you look into the technical details of what you are looking at, you are trying to interchange two limiting operations where it is not allowed. My guess(!) would be that that any value between/including 0 to 1 might turn up with the conditions you are using. It may be that that some stronger conditions, perhaps similar to uniform convergence in some way (or perhaps to ensure that the rates of convergence of Ln/n is in some sense approximately symmetric about 1), might allow you to draw the type of conclusion you want. David Jones === Subject: Re: Markov Chains II Hello Sorry about my english. You are right, practise is an exam or a short test and in media is the expected number of. I considered six differents states E(i) = i quizzes on the desk i = 0..5 and the matrix P : [ 0 1 0 0 0 0 ] [ 1/3 0 2/3 0 0 0 ] [ 1/3 0 0 2/3 0 0 ] [ 1/3 0 0 0 2/3 0 ] [ 1/3 0 0 0 0 2/3 ] [ 1 0 0 0 0 0 ] and I calculated v P = v , v = [ p(0), p(1), ..., p(5) ] with p(0)+p(1)+...+p(5)=1 After that, the solution is 0*p(0) + 1*p(1) + 2*p(2) +...+ 5*p(5) > I'm not sure what I mean by your terminology. > A teacher receive a practise Could this be a quiz? (Quiz = short test) > every morning and put it in a desk. At the > evening, with probability 1/3, he corrects all the practises on the > desk and send it the students. When there are 5 practise on the desk, > he always corrects all of them and send it to the students. > What is, in media, the number of the practises on the desk ? I don't know what in media means. Your answer is a number which is > greater than 1, so I suspect you're asking what the expected number of > quizzes in his desk is, in the long run. (This would require finding > the fixed distribution and then calculating the expected number of > papers in that case.) > My solution is 473 / 292 but I am interested in to compare the > result. I did the calculations but got a different answer. But maybe I'm not > reading the problem correctly. (My answer is 262/211.) > --- Christopher Heckman === Subject: Re: Simply Connected I'm not sure it works, but try this: Let the homotopy J be given by: J(s,t)=H(a,3s) for s<=t/3 J(s,t)=H(p((s-t/3)/(1-2t/3)),t) for t/3<=s<=1-t/3 J(s,t)=H(p,3-3s) for 1-t/3<=s Assuming I haven't messed up, this should be a homotopy relative to {0,1} from the path p to the path q*(constant path at a)*q^(-1), where q is the path given by: q(s)=H(p,s). The path q*(constant path at a)*q^(-1) is evidently a representative of the trivial element of the fundamental group. I think that does it, though please point out any error. -Rotwang > Let S be a contractible space. > How does one show S is simply connected? As S is contractible there's some homotopy H_a or for short, > H in C(SxI,S) with > for all x, H(x,0) = x, H(x,1) = a > where a can be any point in S. That S is path connected is simple. > A path from a to b is, p(t) = H(b,1-t). That S has trivial fundamental group is problem. > Let p be a loop at a. How does one show > p homotopic to p_a relative to { 0,1 }? > p_a is the constant path or loop at a, > p_a(t) = a. That p is homotopic to p_a relative to {0} is immediate for all spaces. > H(s,t) = p(s(1-t)) is a homotopy from p to p_a relative to {0}. What I don't understand is how to find a homotopy from p to p_a relative > to { 0,1 }. There is no problem if S is assumed strongly contractible to > a for all a in S, however there are some spaces, comb spaces in > particular, that though contractible (to a for all a) are not strongly > contractible to a for all a. I suspect the key is the compactness of the domain of p, yet how to use > that, I do not see. === Subject: Re: Simply Connected Two corrections: > I'm not sure it works, but try this: Let the homotopy J be given by: J(s,t)=H(a,3s) for s<=t/3 J(s,t)=H(p((s-t/3)/(1-2t/3)),t) for t/3<=s<=1-t/3 J(s,t)=H(p,3-3s) for 1-t/3<=s Should be J(s,t)=H(a,3-3s) for 1-t/3<=s > Assuming I haven't messed up, this should be a homotopy relative to > {0,1} from the path p to the path q*(constant path at a)*q^(-1), where > q is the path given by: q(s)=H(p,s). Should be q(s)=H(a,s) The path q*(constant path at a)*q^(-1) is evidently a representative of > the trivial element of the fundamental group. I think that does it, > though please point out any error. >-Rotwang === Subject: Re: Simply Connected === Subject: Re: Simply Connected Essential context restored, less we loose track of the details. > As S is contractible there's some homotopy H_a or for short, > H in C(SxI,S) with > for all x, H(x,0) = x, H(x,1) = a > where a can be any point in S. > Let p be a loop at a > p_a is the constant path or loop at a, p_a(t) = a. > I'm not sure it works, but try this: Let the homotopy J be given by: Itworksmuchbetterwithspacesincluded. > J(s,t) = H(a,3s) for s< = t/3 J(s,t) = H(p((s-t/3)/(1-2t/3)), t) for t/3< = s< = 1-t/3 J(s,t) = H(a, 3-3s) for 1-t/3< = s When s = t/3, J(s,t) = H(a,t) J(s,t) = H(p(0),t) = H(a,t) When s = 1 - t/3, J(s,t) = H(p((1 - 2t/3)/(1 - 2t/3), t) = H(p(1),t) = H(a,t) J(s,t) = H(a,t) Checks out. We also have t/3 <= s <= 1 - t/3, which presumes 0 <= 1 - 2t/3, which by golly, is always the case J(s,0) = H(a,0) = a for s <= 0 = H(p(s),0) = p(s) for 0 <= s <= 1 = H(a,0) = a for 1 <= s Ok, J(.,0) = p > J(s,t) = H(a,3s) for s< = t/3 J(s,t) = H(p((s-t/3)/(1-2t/3)), t) for t/3< = s< = 1-t/3 J(s,t) = H(a, 3-3s) for 1-t/3< = s J(s,1) = H(a,3s) for s <= 1/3 = H(p((s - 1/3)/(1/3), 1) = a for 1/3 <= s <= 2/3 = H(a, 3-3s) for 2/3 <= s J(.,1) loop at a reparametrize with f(s) = 0, if s <= 1/3 = 3s - 1, if 1/3 <= s <= 2/3 = 1, if 2/3 <= s J(.,1) o f = p_a; J(.,1) homotopic p_a with respect to { 0,1 }. J(0,t) = H(a,0) = a for 0 <= t = H(p(0),t) = p(0) for t <= 0 = H(a,3) for 1 <= t/3, no! J(1,t) = H(a,3s) for 1 <= t/3, no! = H(p((1 - t/3)/(1 - 2t/3), t) for t/3 <= 1 <= 1 - t/3 = H(p(1),0) = p(1) since t <= 0 = H(a,0) = a for 0 <= t > Assuming I haven't messed up, this should be a homotopy relative to > {0,1} from the path p to the path q*(constant path at a)*q^(-1), where > q is the path given by: q(s) = H(a,s) q is a loop at a and the middle section is constant loop at a. > The path q*(constant path at a)*q^(-1) is evidently a representative of > the trivial element of the fundamental group. I think that does it, > though please point out any error. Ok, well done. What a tour deforce. -- s _______________ | /| | / | | / | t | / | | / | | / | | / | ---------------------------- The middle region is the path p, as deformed by the homotopy H; the s-dependence is compressed so that the entire path fits into the region for each value of t. The left hand region is the path from the fixed point a to the start of the path H(p,t) (using an abuse of notation whose meaning I hope is obvious), given by carrying the point a along the homotopy H. The right hand region is just the inverse path of the left hand region. ---- === Subject: Re: Simply Connected Ok, well done. What a tour deforce. -Rotwang === Subject: Re: Simply Connected Itworksmuchbetterwithspacesincluded. So it does. Will do so in the future. > Ok, well done. What a tour deforce. Whoops, my reparametrization was wrong. Discard it. Even with out it, the homotopy you claimed was correct. === Subject: Re: Simply Connected [a rather opaque series of letters and numbers] Sorry, I should probably attempted a qualitative explanation of how my attempted solution is supposed to work. Will now do so, via the medium of ascii art: s _______________ | /| | / | | / | t | / | | / | | / | | / | ---------------------------- The middle region is the path p, as deformed by the homotopy H; the s-dependence is compressed so that the entire path fits into the region for each value of t. The left hand region is the path from the fixed point a to the start of the path H(p,t) (using an abuse of notation whose meaning I hope is obvious), given by carrying the point a along the homotopy H. The right hand region is just the inverse path of the left hand region. -Rotwang === Subject: Re: Simply Connected If H(x,t) is the homotopy from the identity to the constant map at a , then H(f(s),t) is a homotopy from a loop f to the constant loop at a. It is the composition of maps (s,t) -> (f(s),t) -> H(f(s),t) (map = continuous function) The first part is a map since the projections f(s) and t are maps. > Let S be a contractible space. > How does one show S is simply connected? As S is contractible there's some homotopy H_a or for short, > H in C(SxI,S) with > for all x, H(x,0) = x, H(x,1) = a > where a can be any point in S. That S is path connected is simple. > A path from a to b is, p(t) = H(b,1-t). That S has trivial fundamental group is problem. > Let p be a loop at a. How does one show > p homotopic to p_a relative to { 0,1 }? > p_a is the constant path or loop at a, > p_a(t) = a. That p is homotopic to p_a relative to {0} is immediate for all spaces. > H(s,t) = p(s(1-t)) is a homotopy from p to p_a relative to {0}. What I don't understand is how to find a homotopy from p to p_a relative > to { 0,1 }. There is no problem if S is assumed strongly contractible to > a for all a in S, however there are some spaces, comb spaces in > particular, that though contractible (to a for all a) are not strongly > contractible to a for all a. I suspect the key is the compactness of the domain of p, yet how to use > that, I do not see. === Subject: Re: Find the largest integer a or b without comparisons?? <-dWdnRksJKxdbaLYnZ2dnUVZ_vWdnZ2d@onvoy> Dik T. Winter a .8ecrit : > ... > max(a,b) = ((a+b) >>1) + ((a-b)>>1) > min(a,b) = ((a+b) >>1) - ((a-b)>>1) > > Only works when a > b already where it would be redundant. And a and b must either be both odd or both even. Consider: > max(3, 2) = ((3 + 2) >> 1) + ((3 - 2) >> 1) = > (5 >> 1) + (1 >> 1) = 2 + 0 = 2. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Bonjour, For a and b > 0 we may approximate max(a,b) with formulas such as: (a^ (k+1) + b^ (k+1)) / (a^ k +b^ k) , with large powers of k ; Alain === Subject: Re: Find the largest integer a or b without comparisons?? <6I-dnUiHWOtPeaLYnZ2dnUVZ_tKdnZ2d@onvoy |>> |x| = sqrt(x*x) > |> |>> No comparisons. > | |>You were supposed to use operands +,-,/,*, and MOD. How do you > |>implement sqrt without using any comparison? | There are plenty of ways of computing sqrt without using > | comparisons. Do you think the only way to compute a square root is by > | trial and error? I would like to see four or five ways of computing squart root without > using comparisonsn. Could you show me ? ___________________Gerard S. Why do you want four or five? One would be enough. And you can assume that we only need to compute sqrt(x) where x is a perfect square. But I still do not see how to do it! Could somebody demonstrate in detail how to compute sqrt(4) without comparisons? Derek Holt. === Subject: Re: Find the largest integer a or b without comparisons?? <6I-dnUiHWOtPeaLYnZ2dnUVZ_tKdnZ2d@onvoy> <453FEDD2.5020108@yahoo.it> |>> |x| = sqrt(x*x) > |>|> |>|>> No comparisons. |>|>You were supposed to use operands +,-,/,*, and MOD. How do you > |>|>implement sqrt without using any comparison? > | |>| There are plenty of ways of computing sqrt without using > |>| comparisons. Do you think the only way to compute a square root is by > |>| trial and error? |> I would like to see four or five ways of computing squart root without > |> using comparisonsn. Could you show me ? ___________________Gerard S. > | 1) > | http://www.fourmilab.ch/babbage/library.html > | The square root of argument N is calculated by choosing an initial > | guess of x0=N/2, then using the Newton-Raphson method to refine this > | initial value to the required precision. > | > | Equations for calculating sqrt by Newton-Raphson iteration > | > | x_0=N/2 > | x_{k+1}=(1/2)(N/x_k + x_k) > | ... > | x_{k+1}simeq sqrt(N) > | > | The number of iterations required depends upon the number of decimal > | places required; convergence is quadratic, so relatively few iterations > | suffice. Well, given two numbers, say, 5 and 7, would 1 decimal place (accuracy) > be ok? | 2) Taylor-McLaurin Series, truncated. > | Sqrt(1+x)=1+(1/2)x-(1/8)x^2+(1/16)x^3+...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+.. . > | > | Ok, the previous 1) and 2) are only approximated, ... Ok, so you have two approximations. Two to me, is not many. It looks like > the number of iterations is stopped when the accuracy is close enough to > what one wants, in other words, you do a comparison (to the last > computation). __________________________________________________Gerard S. Also, you need a comparision to determine when to stop ... --- Christopher Heckman === Subject: Re: Find the largest integer a or b without comparisons?? Proginoskes ha scritto: >> |>|>> |x| = sqrt(x*x) >> |>|>> |>|>> No comparisons. >> |>|>You were supposed to use operands +,-,/,*, and MOD. How do you >> |>|>implement sqrt without using any comparison? >> |> |>| There are plenty of ways of computing sqrt without using >> |>| comparisons. Do you think the only way to compute a square root is by >> |>| trial and error? >> |> I would like to see four or five ways of computing squart root without >> |> using comparisonsn. Could you show me ? ___________________Gerard S. >> | 1) >> | http://www.fourmilab.ch/babbage/library.html >> | The square root of argument N is calculated by choosing an initial >> | guess of x0=N/2, then using the Newton-Raphson method to refine this >> | initial value to the required precision. >> | >> | Equations for calculating sqrt by Newton-Raphson iteration >> | >> | x_0=N/2 >> | x_{k+1}=(1/2)(N/x_k + x_k) >> | ... >> | x_{k+1}simeq sqrt(N) >> | >> | The number of iterations required depends upon the number of decimal >> | places required; convergence is quadratic, so relatively few iterations >> | suffice. >> Well, given two numbers, say, 5 and 7, would 1 decimal place (accuracy) >> be ok? >> | 2) Taylor-McLaurin Series, truncated. >> | Sqrt(1+x)=1+(1/2)x-(1/8)x^2+(1/16)x^3+...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+.. . >> | >> | Ok, the previous 1) and 2) are only approximated, ... >> Ok, so you have two approximations. Two to me, is not many. It looks like >> the number of iterations is stopped when the accuracy is close enough to >> what one wants, in other words, you do a comparison (to the last >> computation). __________________________________________________Gerard S. >Also, you need a comparision to determine when to stop ... > --- Christopher Heckman > If you know that a> Proginoskes ha scritto: >> |>|>> |x| = sqrt(x*x) >> |>|>> |>|>> No comparisons. >> |>|>You were supposed to use operands +,-,/,*, and MOD. How do you >> |>|>implement sqrt without using any comparison? >> |> |>| There are plenty of ways of computing sqrt without using >> |>| comparisons. Do you think the only way to compute a square root is by >> |>| trial and error? >> |> I would like to see four or five ways of computing squart root without >> |> using comparisonsn. Could you show me ? ___________________Gerard S. >> | 1) >> | http://www.fourmilab.ch/babbage/library.html >> | The square root of argument N is calculated by choosing an initial >> | guess of x0=N/2, then using the Newton-Raphson method to refine this >> | initial value to the required precision. >> | >> | Equations for calculating sqrt by Newton-Raphson iteration >> | >> | x_0=N/2 >> | x_{k+1}=(1/2)(N/x_k + x_k) >> | ... >> | x_{k+1}simeq sqrt(N) >> | >> | The number of iterations required depends upon the number of decimal >> | places required; convergence is quadratic, so relatively few iterations >> | suffice. >> Well, given two numbers, say, 5 and 7, would 1 decimal place (accuracy) >> be ok? >> | 2) Taylor-McLaurin Series, truncated. >> | Sqrt(1+x)=1+(1/2)x-(1/8)x^2+(1/16)x^3+...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+.. . >> | >> | Ok, the previous 1) and 2) are only approximated, ... >> Ok, so you have two approximations. Two to me, is not many. It looks like >> the number of iterations is stopped when the accuracy is close enough to >> what one wants, in other words, you do a comparison (to the last >> computation). __________________________________________________Gerard S. > Also, you need a comparision to determine when to stop ... >> If you know that a> in N), you can set a stop iteration point. >Once again, you need a comparison to see whether to continue. I'm > assuming you want to truncate a McLaurin Series > >> Sqrt(1+x)=1+(1/2)x-...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+...+((-1)^M(M-3/2)!/( ( -3/2)!M!)) >which translates into >s = 0; > for i = 1 to M > s = s + [...] >and the for loop itself requires comparisons. >This applies to any loop, if you are doing a non-constant number of > iterations. No. You can set the stop point a priori. If you know a priori that a> But I said ... are only approximated ... >And that's another kettle of worms ... er, a can of fish. ... Er, > another problem. > --- Christopher Heckman > Excuse me, but I can't understand what you mean with er, or another kettle of worms and a can of fish. It is a problem, but not another problem. But if you are prepared to pay that price (the approximation) to come to a result... in N, i.e. a and b non negatives. But this problem could be easily overcome if a and b belong to an interval. === Subject: Re: Find the largest integer a or b without comparisons?? Proginoskes ha scritto: >> Proginoskes ha scritto: >> |>|>> |x| = sqrt(x*x) >> |>|>> |>|>> No comparisons. >> |>|>You were supposed to use operands +,-,/,*, and MOD. How do you >> |>|>implement sqrt without using any comparison? >> |> |>| There are plenty of ways of computing sqrt without using >> |>| comparisons. Do you think the only way to compute a square root is by >> |>| trial and error? >> |> I would like to see four or five ways of computing squart root without >> |> using comparisonsn. Could you show me ? ___________________Gerard S. >> | 1) >> | http://www.fourmilab.ch/babbage/library.html >> | The square root of argument N is calculated by choosing an initial >> | guess of x0=N/2, then using the Newton-Raphson method to refine this >> | initial value to the required precision. >> | >> | Equations for calculating sqrt by Newton-Raphson iteration >> | >> | x_0=N/2 >> | x_{k+1}=(1/2)(N/x_k + x_k) >> | ... >> | x_{k+1}simeq sqrt(N) >> | >> | The number of iterations required depends upon the number of decimal >> | places required; convergence is quadratic, so relatively few iterations >> | suffice. >> Well, given two numbers, say, 5 and 7, would 1 decimal place (accuracy) >> be ok? >> | 2) Taylor-McLaurin Series, truncated. >> | Sqrt(1+x)=1+(1/2)x-(1/8)x^2+(1/16)x^3+...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+.. . >> | >> | Ok, the previous 1) and 2) are only approximated, ... >> Ok, so you have two approximations. Two to me, is not many. It looks like >> the number of iterations is stopped when the accuracy is close enough to >> what one wants, in other words, you do a comparison (to the last >> computation). __________________________________________________Gerard S. > Also, you need a comparision to determine when to stop ... >> If you know that a> in N), you can set a stop iteration point. >Once again, you need a comparison to see whether to continue. I'm > assuming you want to truncate a McLaurin Series > >> Sqrt(1+x)=1+(1/2)x-...+((-1)^i(i-3/2)!/((-3/2)!i!))x^i+...+((-1)^M(M-3/2)!/( ( -3/2)!M!)) >which translates into >s = 0; > for i = 1 to M > s = s + [...] >and the for loop itself requires comparisons. >This applies to any loop, if you are doing a non-constant number of > iterations. No. You can set the stop point a priori. If you know a priori that a> But I said ... are only approximated ... >And that's another kettle of worms ... er, a can of fish. ... Er, > another problem. > --- Christopher Heckman > Excuse me, but I can't understand what you mean with er, or another kettle of worms and a can of fish. It is a problem, but not another problem. But if you are prepared to pay that price (the approximation) to come to a result... in N, i.e. a and b non negatives. But this problem could be easily overcome if a and b belong to an interval. === Subject: Re: Find the largest integer a or b without comparisons?? Given a and b, constructing ((a+b) + |a-b|)/2 does not require any >comparisons. > |x|=x if x>=0, > |x|=-x if x<0. > One comparison :). |x| = sqrt(x^2), no comparisons. But SQRT is not on the list of functions that are allowed: +, -, *, /, MOD. (Read the whole thread.) --- Christopher Heckman === Subject: Re: Find the largest integer a or b without comparisons?? >>If x and y are positive integers, try >>x + y - (x mod y) - (y mod x) >> + ((x mod y) mod x) + ((y mod x) mod y) >> - ((x mod y) - ((x-1) mod y)-1) *((y mod x) - ((y-1) mod x)-1)/x >> Let that be F(x,y). Then for any integers u and v, try >> F(u^2 + v^2 + u + 1, u^2 + v^2 + v + 1) - u^2 - v^2 - 1 >> Robert Israel israel@math.ubc.ca >> Department of Mathematics http://www.math.ubc.ca/~israel >> University of British Columbia Vancouver, BC, Canada Hi Robert, Why does that work? What are you talking about? When they're coprime the x mod y mod x, for example 5 and 2 goes 5 2 2 >but 6 and 3 goes 6 3 6. Then you have some product there, we can plug in the other formula, >have min(x,y) + max(x,y) = x + y, the product is the x mod y and x-1 >mod y, minus one, the difference of those, times vice versa for x and >y. Then that's F(x,y), you're saying that's max(x,y) for positive integer >x, y, and it only uses mods except for the divide by x at the >end, what >the hell is that doing there? What's the point? I started by looking at x mod y, y mod x, (x mod y) mod x and (y mod x) mod y, and got something that worked except when x=y. The last line is to correct that case. Case 1) Suppose 0 < x < y. Write y = q x + r where q >= 0, 0 <= r < x. Then x mod y = x, y mod x = r, (x mod y) mod x = x mod x = 0, (y mod x) mod y = r mod y = r, (x mod y) - ((x-1) mod y)-1 = x - (x-1) - 1 = 0, so F(x,y) = x + y - x - r + 0 + r + 0 = y Case 2) Suppose 0 < y < x. By symmetry, we get F(x,y) = x. Case 3) Suppose 0 < x = y. Then x mod y = y mod x = 0, (x mod y) mod x = (y mod x) mod y = 0, (x mod y) - ((x-1) mod y) - 1 = 0 - (x-1) - 1 = -x, (y mod x) - ((y-1) mod x) - 1 = -x as well, and F(x,y) = x + x - 0 - 0 + 0 + 0 -((-x)*(-x))/x = x. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Solve using logs <4qbtfcFmhg4jU1@individual.net > 2(3^(2x)) + 3(3^x) + 2 = 0 > ----- > * Substituting y=3^x gives: > 2y^2 + 3y + 2 = 0 > * which has no (real) solution. Indeed! > So if 3^x is not a real number, how can x exist? For each real number _x_, 2(3^(2x)) + 3(3^x) + 2 > 0. Therefore, > the solution has no real solutions. Could it be that your girlfriend is allowed to look for _complex_ > solutions? > Could be, though I doubt it since this is supposed to be an introduction to logs. === Subject: Re: Solve using logs There is no unibersity that at the start up for a pure math degree would ask for a taking lograithm of complex numbers let alone an Engineering Degree. The chances are there is a misprint and the roots are meant to be real, or the question is a put there for laughs, at 3rd year honours maths people learn about complex logarithms. But anyway, to take the logaritm of a complex number, look at this: http://planetmath.org/encyclopedia/ComplexLogarithm.html HTH > My girlfriend is starting an engineering degree. I foolishly told her I > would help with her maths homework. She is having difficulty > understanding logs. I'm in over my head guys, unless this question doesn't have a solution. > In which case I will feel vindicated. The problem is thus: Solve for x, using logs: 2(3^(2x)) + 3(3^x) + 2 = 0 ----- * Substituting y=3^x gives: 2y^2 + 3y + 2 = 0 * which has no (real) solution. So if 3^x is not a real number, how can x exist? > * Or for pointing out my idiotic mistakes :) === Subject: Re: Solve using logs > There is no unibersity that at the start up for a pure > math degree would ask for a taking lograithm of complex > numbers let alone an Engineering Degree. This is done in Caltech's first year courses, and I'd imagine it's done in many honors calculus courses. In fact, I covered this topic (formally, after Taylor series expansions) several times in high school calculus classes, although this was at a special admissions math/science high school. But in most cases, you're correct. Dave L. Renfro === Subject: Re: Solve using logs My girlfriend is starting an engineering degree. I foolishly told her I >would help with her maths homework. So... you lied to your girlfriend, and now you want us to save your > neck? > LOL, pretty much. >She is having difficulty >understanding logs. And so do you, apparently? > If I'm wrong and this problem does have a solution, then yes I suppose I do. Anyway I got an A at A'Level so I am better qualified than her. 3 years of CS at university has taken it's toll on my mathematic ability. ;) >* Substituting y=3^x gives: >2y^2 + 3y + 2 = 0 >* which has no (real) solution. >So if 3^x is not a real number, how can x exist? Well... if no x exists, then you have solved the problem, have you > not? > Yeah, I'm just not confident enough in my own ability to make that call. So thought I'd check my reasoning with some real experts. === Subject: looking for f,g where f^n + g^n = 1 Hi All I am trying to find out about functions such that f^n + g^n = 1 for n=2 sin and cos are good candiates, but what is known for n<>2 ? I have assumed such functions exists as power series but trying to solve their coefficients for n=3 have proven too much fo me. I assumed f(0)=1 and g(0)=0 but I need more restrictions(?). Any refrences on this topic or any suggestions what topic I should look up,please. Been through any functional equation source I could find but no mentoin of such problems. === Subject: Re: Hello!Every mathmatics wizard I think it is be mixed pair,and I know the function now.Anyhow,thank you very much > Hello everybody! > I am puzzling with a function question,plz help me. > The question is that: > X={1,3,6,2,5} > Y={4,8,2,9,3} > Z={3,6,4,6,4} > does the function f(x,y)=Z exist? What's this? > > Han What do you mean by f(x,y)? Can I take any object from X and from Y and > get any object from Z? Or they need to be mixed pair wise: for example > f(1,4) = 3, f(3,8) = 6 ... etc Paulo Matos === Subject: Sudoku online I have just received an email about what appears to be an online Sudoku puzzle that has js code for installing into random webpages. The author asked me to include it into my webpages as well, but since I don't advertise for free, I thought I might pass this along for people who might want to include it into their webpages. http://www.sudogo/sudoku/sudoku_for_your_website.php Cheerio, -- ------- Gossip is when you hear something you like about someone you don't. === Subject: Re: Ordinal: Definitions > and you have not shown that Ex x = {x 0} contradicts the axiom of >extensionality. > MoeBlee > OK, I failed to proove it, so what is the prove that x=/= {x} without > the axiom of regularity. > Zuhair MoeBlee has already pointed out more than once that you can't prove > it without regularity. On the contrary he said the opposite at least for the case when x = { }. he said exactly that we don't need the axiom of regularity to prove that 0=/= {0}. but I don't know what is his opinion about x=/= {x} when x is not empty. so read well Zuhair === Subject: Re: Ordinal: Definitions > I know all of that. But what I do not know is how do the standard > definition of ordinals you've mentioned do not depend on the axiom of > regularity? The definition I mentioned has built in that every ordinal is well > ordered by membership, so the axiom of regularity is not needed to > ensure that every ordinal is well ordered by membership. Your > definition needs the axiom of regularity to ensure that every ordinal > is well ordered by membership. I've said this about a dozen times > already. > It appears to me that it is only axiom of regularity that can tell > that {} is different form > { {} } No, we don't need the axiom of regularity to prove that 0 not= {0}. MoeBlee Yea but you need the axiom of regularity to prove that for every non empty set x then x=/= {x}. I agree with you that regarding the case of 0=/={0} we do not need the axiom of regularity. This can simply prooved without such an axiom. x={x} / x= { } -> { } = { {} } which violates the axiom of extentiality and the axiom of the empty set. But how can one prove that x=/= {x} without the axiom of regularity . that is my basic question. Can that be proved. The only idea that I have in mind is the following for non empty x . x = {x} <-> x = { {x} } <-> xx = { {x}x } <-> { } = { { } } which viloate the axiom of regularity. But I am still not sure of this line, I think there is something wrong out there. I will repeat my question again. for every non empty set x , do we need the axiom of regularity to prove that x=/= {x} , or this is a theorum that can be proved without this axiom. Zuhair === Subject: Re: Sets and common border >> Then you can use a convention, like having all intervals be open at the >> left and closed on the right. Of course, then, to cover the real line >> you end up including either oo or -oo. >You don't include points oo or -oo in as members of any member of a > partition of the set of real numbers. And you certainly CAN have a > partition of the set of real numbers in which each member is closed at > the right. Not with a last member. >If you want to talk about a different problem, then go ahead, but you > are just adding incorrect statements when you refer to including oo or > -oo in the set of real numbers. I'm trying to satisfy Venkat's need for symmetry. >MoeBlee > === Subject: Re: Middle third characterization by induction of the Cantor ternary set On 26 Oct 2006 09:47:05 -0700, jennifer that have ternary expansion for which a_n is never 1.(If x has >two ternary expansions, we put x in the cantor set if one of the >expansions has no term equal to 1). I want to prove the middle thirds characterization by induction, >starting with x not in C and using x= summation n=1 to oo (a_n/ 3^n) Defining A_n inductively by A_o = [0,1] and A_n+1 is obtained from A_n >by deleting middle open thirds of each interval of A_n. Where do i go from here please and how can i show this induction >explicitely. I know that C is obtained by first removing the middle third (1/3, >2/3) from [0,1], and then removing the middle thirds (1/9, 2/9) and >(7/9,8/9) of the remaning intervals and so on. Show that the middle-thirds set and the set of all numbers with ternary expansions of the specified type are compact sets which satisfy A = (1/3) A union (2/3 + 1/3 A). Now show that A -> (1/3) A union (2/3 + 1/3 A) is a strict contration in the Hausdorff metric on compact sets, hence it has exactly one fixed point. proof you're looking for. Let me know when the homework is due - I'll try to post it the next day...) David C. Ullrich === Subject: Re: algebra with Z[i]/<2+2i>. > hello sir~ >i want to find the elements of Z[i]/<2+2i>. >i think... >let a+bi in Z[i]. >a+bi = a(2+2i) - a + (b-2a)i. >and b-2a = 4k, 4k+1, 4k+2, 4k+3 form. > (k is integer) >and (1+i)(2+2i) = 2+2i+2i-2 = 4i (*). >since a(2+2i) in <2+2i>, > if b-2a = 4k form, > a+bi = a(2+2i) - a + (b-2a)i in -a+<2+2i>. > because, (b-2a)i = 4ki in <2+2i> by (*). >if b-2a = 4k+1 form, > a+bi = a(2+2i) - a + (b-2a)i in -a+1+<2+2i>. >if b-2a = 4k+2 form, > a+bi = a(2+2i) - a + (b-2a)i in -a+2+<2+2i>. >if b-2a = 4k+3 form, > a+bi = a(2+2i) - a + (b-2a)i in -a+3+<2+2i>. >so, Z[i]/<2+2i> = {a + <2+2i> | a in Z}. >but we know the next..... > c+<2+2i> = d+<2+2i>. > <=> (c-d) in <2+2i>. > and (1-i)(2+2i) = 2+2i-2i+2 = 4. >so, if c-d = 4, then c+<2+2i> = d+<2+2i>. > so, example) 1+<2+2i> = 5+<2+2i> = .... >it means that > Z[i]/<2+2i> = {a + <2+2i> | a = 0,1,2,3}. >my thinking is right ?? > I am not sure whether your result is right, since if you consider the surjection Z[i] -> Z mapping i to -1, then what is your quotient in terms of a quotient of Z? HTH. J. === Subject: Re: algebra with Z[i]/<2+2i>. > it means that > Z[i]/<2+2i> = {a +ci+ <2+2i> | a = 0,1,2,3, c=0,1,2,3}. >is this no problem ?? I didn't go through all of your solution. But the pairs (a,c)=(0,0) and (2,2) induce the same coset. So how many element does the quotient have? J. === Subject: Re: algebra with Z[i]/<2+2i>. > it means that > Z[i]/<2+2i> = {a +ci+ <2+2i> | a = 0,1,2,3, c=0,1,2,3}. > is this no problem ?? I didn't go through all of your solution. But the pairs (a,c)=(0,0) and > (2,2) induce the same coset. So how many element does the quotient have? yes. you're right. um......i think... a +ci+ <2+2i> = (a+2)+(c+2)i+<2+2i>. so, (a,c) = (0,0) = (2,2) (0,1) = (2,3) (0,2) (0,3) (1,0) = (3,2) (1,1) = (3,3) (1,2) (1,3) (2,1) (3,0) (3,1) thus, how many element does the quotient have? 11. but, it's wrong. because, 1+2i+<2+2i> = 3+0i+<2+2i> <=> -2+2i in <2+2i>. very complex.... anyway, a+ci+<2+2i> = (a+2)+(c-2)i+<2+2i>. and a+ci+<2+2i>=(a-2)+(c+2)i+<2+2i>. because, 2-2i in <2+2i> maybe, i think.... a+ci+<2+2i> = (a+2)+(c+2)i +<2+2i>. a+ci+<2+2i> = (a-2)+(c-2)i +<2+2i>. a+ci+<2+2i> = (a+2)+(c-2)i +<2+2i>. a+ci+<2+2i> = (a-2)+(c+2)i +<2+2i>. so, (a,c) = (0,0) = (2,2) (0,1) = (2,3) (0,2) = (2,0) (0,3) = (2,1) (1,0) = (3,2) (1,1) = (3,3) (1,2) = (3,0) (1,3) = (3,1) thus, how many element does the quotient have? 8. how do you think about it ? === Subject: Re: algebra with Z[i]/<2+2i>. >thus, how many element does the quotient have? 8. >how do you think about it ? That sounds good! Here is a reasoning why 8 is correct. Note that Z[i] can be represented by Z[X]/(X^2+1) which is used in the following. The surjection Z[i]/2(1+i)Z[i] -> Z[i]/(1+i)Z[i] has kernel K:=(1+i)Z[i]/2(1+i)Z[i]. K is isomorphic to K':=Z[i]/2Z[i], since 1+i is not a zero-divisor in Z[i] (which is an integral domain). Then K' = Z[X]/(2,X^2+1) = (Z/2Z)[X]/(X^2+1) (= means isomorphism -at least of groups- here). Hence K' is a vector space over Z/2Z with basis {1,X}, hence it has 4 elements. Now Z[i]/(1+i)Z[i] = Z[X]/(1+X,X^2+1) =(*) Z/(1^2+1)Z = Z/2Z where X is mapped to -1 in (*) (here = means isomorphism again). THus the latter group has two elements. This implies that Z[i]/2(1+i)Z[i] has 2*4=8 elements. J. === Subject: Re: algebra with Z[i]/<2+2i>. > thus, how many element does the quotient have? 8. > how do you think about it ? That sounds good! Here is a reasoning why 8 is correct. Note that Z[i] can be represented by Z[X]/(X^2+1) which is used in the > following. The surjection Z[i]/2(1+i)Z[i] -> Z[i]/(1+i)Z[i] has kernel > K:=(1+i)Z[i]/2(1+i)Z[i]. K is isomorphic to K':=Z[i]/2Z[i], since 1+i is not a zero-divisor in > Z[i] (which is an integral domain). Then K' = Z[X]/(2,X^2+1) = > (Z/2Z)[X]/(X^2+1) (= means isomorphism -at least of groups- here). Hence > K' is a vector space over Z/2Z with basis {1,X}, hence it has 4 elements. Now Z[i]/(1+i)Z[i] = Z[X]/(1+X,X^2+1) =(*) Z/(1^2+1)Z = Z/2Z where X is > mapped to -1 in (*) (here = means isomorphism again). THus the latter > group has two elements. This implies that Z[i]/2(1+i)Z[i] has 2*4=8 elements. oh...very hard. i will try to understand your advice. thank you very much always. === Subject: Re: algebra with Z[i]/<2+2i>. <454193B0.2060502@web.de> <4541DC97.7010300@web.de> thus, how many element does the quotient have? 8. > how do you think about it ? > That sounds good! Here is a reasoning why 8 is correct. > Note that Z[i] can be represented by Z[X]/(X^2+1) which is used in the > following. > The surjection Z[i]/2(1+i)Z[i] -> Z[i]/(1+i)Z[i] has kernel > K:=(1+i)Z[i]/2(1+i)Z[i]. > K is isomorphic to K':=Z[i]/2Z[i], since 1+i is not a zero-divisor in > Z[i] (which is an integral domain). Then K' = Z[X]/(2,X^2+1) = > (Z/2Z)[X]/(X^2+1) (= means isomorphism -at least of groups- here). Hence > K' is a vector space over Z/2Z with basis {1,X}, hence it has 4 elements. > Now Z[i]/(1+i)Z[i] = Z[X]/(1+X,X^2+1) =(*) Z/(1^2+1)Z = Z/2Z where X is > mapped to -1 in (*) (here = means isomorphism again). THus the latter > group has two elements. > This implies that Z[i]/2(1+i)Z[i] has 2*4=8 elements. oh...very hard. > i will try to understand your advice. > thank you very much always. Here's a clue: what the the ideals in Z[i] containing <2+2i>? They will be in one-to-one correspondence with the ideals in Z[i]/<2+2i>. === Subject: Re: Strange Borel quote >I have had a look at Les Paradoxes de L'infini and (...) >... > Although I haven't read all of the book, I don't > think Sazanov's quote is to be found in this book. Same with me. The Google-books quote didn't contain it either. So the mystery remains, for now. -- Herman Jurjus === Subject: Re: LCM equation?- I don't know why. It may even not be true. Try to take advantage from GCD (a, b) * LCM (a, b) = a * b Perhaps this will help. Johan E. Mebius >GCD(a,b) is well known to satisfy linear Diophantine equation ax + by = GCD(a,b) What about LCM, apparently there is no linear Diophantine equation it >is accociated with. Why? > === Subject: i is not in Q(2^(1/4) i) show that i is not in the field Q(2^(1/4)i) where Q is the rational numbers. I did it by brute force. is there an easier way to show this? === Subject: Re: i is not in Q(2^(1/4) i) > show that i is not in the field Q(2^(1/4)i) where Q is the rational > numbers. > I did it by brute force. is there an easier way to show this? Assume that 2^(1/4) be *real*. If i was in K:=Q(2^(1/4)i), then (?) 2^(1/4) would be zero of a polynomial over Q(i) of degree 2, hence a zero of a polynomial over Q of degree 2. Contradiction to the irreducibility of X^4-2 in Q[X]. HTH. J. === Subject: Re: i is not in Q(2^(1/4) i) Hi Christopher, let me fill in some bits of my sketchy proof. > show that i is not in the field Q(2^(1/4)i) where Q is the rational > numbers. > I did it by brute force. is there an easier way to show this? >> Assume that 2^(1/4) be *real*. If i was in K:=Q(2^(1/4)i), then (?) >> 2^(1/4) would be zero of a polynomial over Q(i) of degree 2, >How do you deduce this without having to resort to a brute force? The > degree of any of the field extensions (except Q(2^(1/4)*i) over Q) is > certainly at most 2, but you have to use brute force to get rid of the > case where the extension has degree 1. If x:=2^(1/4) was in Q(i), then Q(x) would be equal to Q(i), hence x would be algebraic over Q of degree 2 which contradicts the fact that X^4-2 is the minimal polynomial of x over Q. > Perhaps you should say ... of degree _at most_ 2, just to be safe > here. > >> hence a zero of a polynomial over Q of degree 2. >I'm even less sure of this. Couldn't the degree double here, since the > degree of Q(i) over Q is 2? Have a close look at the minimal polynomial of x over Q(i): Taking real parts of its coefficients yields a polynomial of degree 2 over Q having the zero x (use that x be real). > --- Christopher Heckman > >> Contradiction to the irreducibility of X^4-2 in Q[X]. >> HTH. >> J. > Or am I missing something here? J. === Subject: Re: Math field, corrupted in late 1800's > The math field was corrupted in that late 1800's in number theory, when > some rather intriguing mistakes were made leading to the acceptance of > ideal theory. The unfortunate response of mathematicians at the time was to declare > the area pure and talk as if non-practicality were good!!! That gave free rein for the error to propagate and opened the door for > cons, people without consciences who learned how to talk math-ese, and > work together, to maintain that wrong results not proven mathematically > were correct. Over a hundred years later, we have a system that is fully corrupted > and capable of claiming just about anything, so no, Andrew Wiles did > not prove FLT, and you can find all kinds of interesting problems and > mistakes and amazing denial when it comes to important mathematical > arguments, like, um, how many of you know of Plotnikov and his claims > of proving P=NP? After years of effort I just found a way to generalize the factoring > problem. Seems like no one noticed that x^2 - y^2 = 0 mod T was just a first step--a primitive case of a more generalized set of > equations: so add S - 2xk = 0 mod T and you have (x+k)^2 = y^2 + S + k^2 + nT : where you pick S, k and n, and n is just a difference from the other > two equations. from just one number, JSH now wants you to pick or guess what the other x,k,y,n are. what a sick little monkey, put him back on his chain, and give him back to the organ grinder. Now that is just a damn good idea, but you won't hear mathematicians > getting excited about it as they're cons in a corrupt system hoping to > hold on to the lie of their supposed mathematical discoveries against > the reality of mine, and the mistakes that entered into the field in > the late 1800's. Kind of dramatic eh? I wish it were wrong. I kind of put up these equations hoping they are > wrong, so I can focus on my other mathematical research without > worrying about the fate of the world. too late the world is doomed by your equations, ah nuts, I needed to mow the lawn......... Proof against the field is its continuing to ignore this research. > Time is part of my prosecutorial argument. that is self-prosecutorial They knew, they bet the world, and they will lose. But remember, they > put other people's lives on the line, and for what? what line? who's lives? Oh, no! Stock Brokers! sky!! For math lies? yes, it does, 1 + 1 = 10 (except in binary) > === Subject: Re: laplace transform >Somehow I am making a mistake using the Laplace transform. I >am taking >the LT of a function, then finding its inverse using residues. I find >the LT of the function f(t) = 1, 0<= t <=1 and f(t) = 0 for 1F(s) = (1-exp(-s))/s. But the integral for the inverse LT is f(t) = >(1/2*Pi*i) int (-infinity to infinity) exp(s*t) * F(s) ds is using >residues equal to f(t) = 0 for all t. The integrand in the LT is >analytic for all s (it has a removable singularity at s = 0. You can't do the integral by residues unless you can do something about the integral on the return arc, e.g. get it to go to 0. >I am starting to think ... is this a true theorem ? The Laplace >transform of a classical function f(t) is an entire function iff the >function has compact support ... i.e. f(t) = 0 for t> T where T is >some positive number. Not true. For example, exp(-t^2/4) does not have compact support, and its Laplace transform sqrt(pi) exp(s^2) erfc(s) is entire. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: This Week's Finds in Mathematical Physics (Week 240) >> In analysis, X^Y >> can approach anything between 0 and 1 when X and Y approach 0. >If X^Y approaches Z, then X^(-Y) appoaches 1/Z, so the actual range of >possible values is between 0 and infinity, including both endpoints. I was imagining both X and Y being positive, perhaps because I had set theory on my mind, and sets have a nonnegative number of elements. :-) But, of you're right if Y can be negative. Here are some other corrections and comments, which have been used to improve the web version of week240, available here: http://math.ucr.edu/home/baez/week240.html before Dolan and Trimble's that also described the free cartesian closed category on one object in terms of games. I was gently set straight by Dominic Hughes, who has permitted me to attach this post of his from the category theory mailing list: This backtracking game characterisation has been known since around '93-'94, in the work of Hyland and Ong: M. Hyland and L. Ong. On full abstraction for PCF. Information and Computation, Volume 163, pp. 285-408, December 2000. [Under review for 6 years!] ftp://ftplab.ox.ac.uk/pub/Documents/techpapers/Luke.Ong/pcf.ps.gz (PCF is an extension of typed lambda calculus.) My D.Phil. thesis extended the lambda calculus (free CCC) characterisation to second-order, published in: Games and Definability for System F. Logic in Computer Science, 1997 http://boole.stanford.edu/~dominic/papers/ To characterise the free CCC on an *arbitrary* set {Z,Y,X,...} of generators (rather than a single generator, as you discuss), one simply adds the following Copycat Condition: (*) Whenever first player plays an occurrence of X, the second player must play an occurrence of X. [Try it: see how X -> Y -> X has just one winning strategy.] Although the LICS'97 paper cited above appears to be the first place the Copycat Condition appears in print, I like to think it was already understood at the time by people working in the area. Technically speaking, winning strategies correspond to eta-expanded beta-normal forms. See pages 5-7 of my thesis for an informal description of the correspondence. It sounds like you've reached the point of trying to figure out how composition should work. Proving associativity is fiddly. Hyland and Ong give a very elegant treatment, via a larger CCC of games in which *both* players can backtrack. The free CCC subcategory is carved out as the so-called *innocent* strategies. This composition is almost identical to that presented by Coquand in: A semantics of evidence for classical arithmetic. Thierry Coquand. Proceedings of the CLICS workshop, Aarhus, 1992. Dominic PS A game-theoretic characterisation with an entirely different flavour (winning strategies less obviously corresponding to eta-long beta-normal forms) is: Abramsky, S., Jagadeesan, R. and Malacaria, P., Full Abstraction for PCF. Info. & Comp. 163 (2000), 409-470. http://weblab.ox.ac.uk/oucl/work/samson.abramsky/pcf.pdf [Announced concurrently with Hyland-Ong, around '93-'94.] On a different subject, James Dolan had this to say: you describe holodeck strategies for lady or tiger where you take back just when the tiger is about to eat you, but that's not the way it works. you take back just _after_ the tiger has eaten you. (i guess that this is partially because of your lack of experience with computer games with a saved game feature. typically you die in the game and the computer plays some sort of funeral or at least funereal music; then you're taken to the reincarnation gallery where you select one to return to from your catalog of previous lives. or something like that.) Also, in the first version of this Week's Finds I claimed that all systematic ways of picking an element of (X^X)^(X^X) could be defined using the lambda calculus. I was disabused of this notion by Vaughan Pratt, who In week240, you said The moral of this game is that all systematic methods for picking an element of (X^X)^(X^X) for an unknown set X can be written using the lambda calculus. What is unsystematic about the contagious-fixpoint functional? This is the functional that maps those functions that have any fixpoints to the identity function (the function that makes every element a fixpoint) and functions without fixpoints to themselves (thus preserving the absence of fixpoints). It's a perfectly good functional that is equally well defined for all sets X, its statement in no way depends on X, and conceptually the concept of contagious fixpoints is even intuitively natural, but how do you write it using the lambda calculus? Many more examples in this vein at JPAA 128, 33-92 (Pare and Roman, Dinatural numbers, 1998). The above is the case K = {0} of Freyd's (proper) class of examples. Vaughan Here Pratt uses functional to mean what I was calling an operator. Finally, Tom Payne at UCR coaxed me into using more standard lambda calculus notation. ........................................................................ Puzzle 29: Stravinsky's Rite of Spring caused a ruckus in Paris, but which piece actually got him *arrested* in Boston? If you get stuck, try: http://math.ucr.edu/home/baez/puzzles/ === Subject: Re: Lindelof covering theorem, >>is it that any open cover can be reduced to a countable >>subcover? Yes that is basically the correct statement of the theorem. What does separable mean--that term hasn't been defined in the book (at least not yet). So include what? === Subject: Re: Lindelof covering theorem, > What does separable mean--that term hasn't been defined in the book (at least not yet). In this case separable means that there is a countable dense subset of the metric space. === Subject: R-homomorphisms Just as a linear transformation can be defined by specifying its action on the basis vectors, I am trying to show that an R-homomorphism can be defined by assigning arbitrary values on the elements of a basis (assuming it exists) and extending by linearity. I'm not really sure what I am supposed to show. But this is what I came up with: Let B = {v_i:i in I} be a basis for module M. Specify T(v_i) for all v_i in B. Then extend the domain of T to M using linearity: T(v) = T(a1*v1+...+an*vn) = a1*T(v1)+...+an*T(vn) I then show T is indeed a homomorphism (basically linear). Then I suppose T(v_i) = Q(v_i) Let u = a1*v1 + ... + an*vn T(u) = a1*T(v1)+...+an*T(vn) Q(u) = a1*Q(v1)+...+an*Q(vn) Thus T = Q, so it is unique. Is that correct/all I need to do? === Subject: R-homomorphisms Just as a linear transformation can be defined by specifying its action on the basis vectors, I am trying to show that an R-homomorphism can be defined by assigning arbitrary values on the elements of a basis (assuming it exists) and extending by linearity. I'm not really sure what I am supposed to show. But this is what I came up with: Let B = {v_i:i in I} be a basis for module M. Specify T(v_i) for all v_i in B. Then extend the domain of T to M using linearity: T(v) = T(a1*v1+...+an*vn) = a1*T(v1)+...+an*T(vn) I then show T is indeed a homomorphism (basically linear). Then I suppose T(v_i) = Q(v_i) Let u = a1*v1 + ... + an*vn T(u) = a1*T(v1)+...+an*T(vn) Q(u) = a1*Q(v1)+...+an*Q(vn) Thus T = Q, so it is unique. Is that correct/all I need to do? === Subject: R-homomorphisms Just as a linear transformation can be defined by specifying its action on the basis vectors, I am trying to show that an R-homomorphism can be defined by assigning arbitrary values on the elements of a basis (assuming it exists) and extending by linearity. I'm not really sure what I am supposed to show. But this is what I came up with: Let B = {v_i:i in I} be a basis for module M. Specify T(v_i) for all v_i in B. Then extend the domain of T to M using linearity: T(v) = T(a1*v1+...+an*vn) = a1*T(v1)+...+an*T(vn) I then show T is indeed a homomorphism (basically linear). Then I suppose T(v_i) = Q(v_i) Let u = a1*v1 + ... + an*vn T(u) = a1*T(v1)+...+an*T(vn) Q(u) = a1*Q(v1)+...+an*Q(vn) Thus T = Q, so it is unique. Is that correct/all I need to do? === Subject: Re: R-homomorphisms > Just as a linear transformation can be defined by specifying its action on the basis vectors, I am trying to show that an R-homomorphism can be defined by assigning arbitrary values on the elements of a basis (assuming it exists) and extending by linearity. I'm not really sure what I am supposed to show. But this is what I came up with: Let B = {v_i:i in I} be a basis for module M. Specify T(v_i) for all v_i in B. Then extend the domain of T to M using linearity: T(v) = T(a1*v1+...+an*vn) = a1*T(v1)+...+an*T(vn) I then show T is indeed a homomorphism (basically linear). Then I suppose T(v_i) = Q(v_i) Let u = a1*v1 + ... + an*vn T(u) = a1*T(v1)+...+an*T(vn) > Q(u) = a1*Q(v1)+...+an*Q(vn) Thus T = Q, so it is unique. Is that correct/all I need to do? > Yes, it is right. For infinite sums to make sense you have to define some notion of convergence. If a set is a basis in the algebraic sense, then every element is a finite linear combination of some members of the set. Your argument is fine. === Subject: Re: modifying mahalanobis distance <26025870.1161861961173.JavaMail.jakarta@nitrogen.mathforum.org Error is assumed to be heterogenous, in fact, I assume that every data point has a unique error for maximum generality. The distribution of all objects is assume to be a mixture of Gaussians. I am trying to modify the standard EM clustering algorithm (specifically the E step) to give greater weights to data points with small errors and smaller weights to data points with large errors. So, as I understand it, in the errorless case you want to solve for (p_j, m_j, K_j), j = 1,...,J, to maximize prod_i sum_j (p_j L[y_i|m_j,K_j]), where i indexes objects, j indexes clusters, y_i is a data vector, p_j is a mixing proportion, m_j is a mean vector, K_j is a covariance matrix and L is the normal likelihood function; and when the measurements have error you would (ideally) replace L[y_i|m_j, K_j] by int_x L[x|m_j,K_j]*L[y_i|x,C_i], where C_i is the covariance matrix of the errors for object i. The goal is to find a way to avoid integrating over x without creating too much error of approximation. If that's wrong then please correct me. And in any case, where are you getting the C_i matrices? === Subject: Re: modifying mahalanobis distance > L[y_i|m_j, K_j] by int_x L[x|m_j,K_j]*L[y_i|x,C_i], >where C_i is the covariance matrix of the errors for > object i. > > The goal is to find a way to avoid integrating over x > without creating too much error of approximation. That looks about right. I am basically for a way of modifying the standard EM algorithm to achieve this. > And in any case, where are you getting the C_i > matrices? > C_i is estimated using non parametric regression method and technically it is the uncertainty about the prediction. === Subject: Re: Forbidden graph structures? > I have a question regarding complete, asymmetric directed graphs (also > known as 'tournaments') that is really bothering me: > Consider a strongly connected tournament, G [so that for every pair of > vertices (x,y), > either x 'beats' y (x->y) or y beats x (y->x), but not both, and not > neither. Strongly connected implies there exists a directed path from > any vertex to any other vertex. ] So there are a total of N(N-1)/2 > edges in this graph. > I want to show that the following is impossible: > (1) > for all vertices x, for each y such that y->x, x beats two things that > beat y. > that is: if y->x, then exits w, w' (w' ne w) such that x->w->y > and x->w'->y. > The 'reason' I think this is impossible is: if you try to > construct such a graph, you always end up with 'too many edges'. I > tried to formalize this by counting up how many edges (1) would imply, > but I keep over counting. Is is possible to satisfy (1)? >Yes. Here is an example with 7 vertices a,b,c,d,e,f,g. >a beats c,f,g. > b beats a,e,f. > c beats b,e,g. > d beats a,b,c. > e beats a,d,g. > f beats c,d,e. > g beats b,d,f. No dount there's a projective plane hiding in there somewhere. Or quadratic residues modulo 7. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Forbidden graph structures? I have a question regarding complete, asymmetric directed graphs (also >known as 'tournaments') that is really bothering me: > Consider a strongly connected tournament, G [so that for every pair of >vertices (x,y), >either x 'beats' y (x->y) or y beats x (y->x), but not both, and not >neither. Strongly connected implies there exists a directed path from >any vertex to any other vertex. ] So there are a total of N(N-1)/2 >edges in this graph. > I want to show that the following is impossible: >(1) >for all vertices x, for each y such that y->x, x beats two things that >beat y. > that is: if y->x, then exits w, w' (w' ne w) such that x->w->y >and x->w'->y. > The 'reason' I think this is impossible is: if you try to >construct such a graph, you always end up with 'too many edges'. I >tried to formalize this by counting up how many edges (1) would imply, >but I keep over counting. Is is possible to satisfy (1)? > Yes. Here is an example with 7 vertices a,b,c,d,e,f,g. > a beats c,f,g. > b beats a,e,f. > c beats b,e,g. > d beats a,b,c. > e beats a,d,g. > f beats c,d,e. > g beats b,d,f. No dount there's a projective plane hiding in there somewhere. > Or quadratic residues modulo 7. No doubt, and if you can explain it to me, I'd appreciate it! I couldn't see how to do anything with quadratic residues, so I decided to try using 7 vertices and making the out-neighborhoods a Steiner triple system. So I had to set up a 1-1 correspondence between the Steiner triples and the points so that the resulting digraph would be a tournament. This was easy enough to do, but my method was pure trial-and-error. Can you explain this correspondence in terms of projective geometry? === Subject: Re: help: a Combinatorial problem ... > N+1 nodes are deployed along a straight line. The indices of nodes are > 1,2,..., N+1. The 1st node is at the left end of the line and the > (N+1)th node is at the right end of the line. We call nodes 1,...,N > as transmitters because each of these node has a packet to be sent >to node N+1. [Snip: packet of node i has to go through nodes i+1,...,N in (N-i+1) hops] > (*) At any step, only one out of any three neighboring nodes can send a > packet to the node on its right-hand side if it has a packet to send. Parallel transmissions of packets can be scheduled in one step as long > as (*) is satisfied. > [Snip: How to find optimal schedule to minimize total steps for all packets to get to N+1 and how to prove it optimal] d is sending, then b,c,e,f cannot be sending, but a and g could be. N/3 packet sends per step. Also, N*(N+1)/2 sends are needed to send all the packets to node N+1. Hence a lower bound of steps for any algorithm is given by (N*(N+1)/2)/(N/3) = 3(N+1)/2 steps. This lower bound might not be useful, being only half as much as the 3N steps the following apparently-optimal method takes: At step i, for all j, if i==j mod 3, let node j send a packet. -jiw === Subject: Re: help: a Combinatorial problem <4541A106.93F3210F@pat7 ... > N+1 nodes are deployed along a straight line. The indices of nodes are > 1,2,..., N+1. The 1st node is at the left end of the line and the > (N+1)th node is at the right end of the line. We call nodes 1,...,N > as transmitters because each of these node has a packet to be sent >to node N+1. > [Snip: packet of node i has to go through nodes i+1,...,N in (N-i+1) hops] > (*) At any step, only one out of any three neighboring nodes can send a > packet to the node on its right-hand side if it has a packet to send. > Parallel transmissions of packets can be scheduled in one step as long > as (*) is satisfied. > [Snip: How to find optimal schedule to minimize total steps for > all packets to get to N+1 and how to prove it optimal] d is sending, then b,c,e,f cannot be sending, but a and g could be. > N/3 packet sends per step. Also, N*(N+1)/2 sends are needed to > send all the packets to node N+1. Hence a lower bound of steps for > any algorithm is given by (N*(N+1)/2)/(N/3) = 3(N+1)/2 steps. > This lower bound might not be useful, being only half as much as > the 3N steps the following apparently-optimal method takes: > At step i, for all j, if i==j mod 3, let node j send a packet. -jiw Yes, initially N/3 packets can be scheduled in each step. But when packets are more and more concentrated on the nodes which are closer to the destination node (node N+1), less than N/3 could be scheduled. The scheme you mentioned might not be optimal as the distribution of packets on nodes becomes non-uniform with the progress of moving packets. Thus, not every node has a packet to transmit at every step, e.g., if you schedule node a to transmit at the 1st step, it has no more requirement for transmission in the following steps. I should say it more clearly - each node only has one packet to send. I hope that we can get the exact number of steps and prove the optimality. Cloud === Subject: Re: Zen and...Math?? >Putting advanced mathematics on the same level as Zen, which is nonsense > and > absurdity made method, is at least hubris. >> Zen is neither nonsense nor sense >That's NONSENSE, right THERE. Would it make sense, or non-sense to ask the color blue that you see is the same color that I see? The answer is whatever color you see is what you see and whatever color I see is what I see. And, as Suzuki alluded to, that's the end of it. If you would like to make sense or not to make sense out of any comparison, that's your issue, but it doesn't alter what you see, or what somebody else sees. >Label N = Nonsense, S = Sense and Omega = Everything (Universe), >The dichotomy nonsense vs sense always exists, because N U S = Omega >To say that Zen is neither nonsense nor sense you are saying Zen not in > Omega. But Zen is SOMETHING (a way, philosophy, whatever), so it is in Omega, > contradiction, therefore Zen in Omega, thus either Zen in S or Zen in N. >Since, as you say, Zen not in Omega, it follows that Zen in N, thus it is > nonsense. You should make up your mind. If you want to talk about mathematics, state the axioms for Omega, U, N, S, ... If you want to talk about Zen, talk about something that neither make sense or non-sense. > === Subject: Re: Zen and...Math?? Nam Nguyen ó.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > Putting advanced mathematics on the same level as Zen, which is nonsense > and > absurdity made method, is at least hubris. >> Zen is neither nonsense nor sense > That's NONSENSE, right THERE. Would it make sense, or non-sense to ask the color blue that you see > is the same color that I see? The answer is whatever color you see is > what you see and whatever color I see is what I see. And, as Suzuki > alluded to, that's the end of it. That's sense. Empirical sense. I see the same color that you see. How do I know? I don't, but empiricism teaches me that most humans agree on color hue validation. Otherwise there would be confusion. > If you would like to make sense > or not to make sense out of any comparison, that's your issue, but it > doesn't alter what you see, or what somebody else sees. Has nothing to do with whether Zen is nonsense or not, though. > Label N = Nonsense, S = Sense and Omega = Everything (Universe), > The dichotomy nonsense vs sense always exists, because N U S = Omega > To say that Zen is neither nonsense nor sense you are saying Zen not in > Omega. But Zen is SOMETHING (a way, philosophy, whatever), so it is in Omega, > contradiction, therefore Zen in Omega, thus either Zen in S or Zen in N. > Since, as you say, Zen not in Omega, it follows that Zen in N, thus it is > nonsense. You should make up your mind. If you want to talk about mathematics, > state the axioms for Omega, U, N, S, ... If you want to talk > about Zen, talk about something that neither make sense or non-sense. I cannot talk about something that neither makes sense nor nonsense, simply because such a thing does not exist. Neither do I need to state axioms for N, S and Omega. It's empirically obvious that anything must be either in S or in N. Something either makes sense or it doesn't. You cannot have neither or both. You claim that Zen is neither, so by that statement, Zen is nonsense. -- ------- Drive defensively. Buy a tank. === Subject: Re: Zen and...Math?? > It's empirically obvious > that anything must be either in S or in N. I makes sense that the adjective autological is autological. It also makes sense that the adjective autological is heterological. On the other hand it doesnÇt make sense that autological can be both autological and heterological at the same time. Looks like an intersection of S and N. > Something > either makes sense or it > doesn't. You cannot have neither or both. You claim > that Zen is neither, so by > that statement, Zen is nonsense. There is far more beyond two-valued logic. Even mathematicians like Brower claim so. === Subject: Re: Zen and...Math?? Am 27.10.2006 09:34 schrieb Nam Nguyen: >You should make up your mind. If you want to talk about mathematics, > state the axioms for Omega, U, N, S, ... If you want to talk > about Zen, talk about something that neither make sense or non-sense. :-) ... Applause... Gottfried Helms === Subject: Re: Zen and...Math?? Zen Buddhism is very much about >> understaning complicated abstract structures, only, typically, where >> the self is concerned -- > It is ? > Hhmmm....I've been reading Suzuki and that's not my understanding of >the situation. > Zen is beyond language and logic actually. > I don't think Zen and mathematics compatable at all. >Zen masters would think mathematics a waste of time. > Masters of anything sometimes think everything outside their own > field is a waste of time. That's an occupational hazard when you > devote your whole life to one thing. > All mathematics is, is description. You attempt to describe >something, its behaviour, shape, motion....its PHYSICAL ATTRIBUTES. > Obviously you don't know much about modern mathematics. Mathematics > can describe physical attributes of something, but it can also describe > things that have very little to do with the physical world. What is > physical about cohomology of infinite-dimensional Hilbert manifolds? > that is an exceedingly poor example from you!! the cohomology of infinite dimensional hilbert manifolds > describes the obstructions to extension of the state space > for physical observables in quantum theory! if i want to model physical reality > and i have a partial quantum model given by some hilbert manifold > (or some characterisation of its principal bundles) > then its cohomology would be one of the first things i'd study > to provide a more complete physical model... the thing is... i am sure you know this! thus i must study the contradiction > as all zen students must... > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > galathaea: prankster, fablist, magician, liar Looks like you solved the koan by realizing it was one. --c === Subject: Re: Probability that the sum of n uniform-(0,1) variables is < 1, is 1/n! , > I recently read that for RVs X_1, X_2, ..., X_n, each uniform on (0,1), > Pr[ sum X_i < 1] = 1/n! >It feels like there must be an easy way to see this (without moment > generating functions, LaPlace transforms, Fourier transforms, > characteristic functions, etc), but it eludes me. Anyone? Plenty of good answers now. You might want to prove Dirichlet's integral. int f(x1+x2+...+xn) x1^a1 x2^a2 ... xn^an dx1/x1 dx2/x2 ...dxn/xn = G(x1)G(x2)...G(xn)/G(x1+x2+...xn) int_0^1 f(t) t^{a1+a2+...+an} dt/t where the first integral is over 0 <= x1,x2,...,xn <= 1, and x1+x2+...+xn <=1. G(x) is the gamma function: G(x) = (x-1)!. Will prove your formula as well as the volume of n-balls. -- Michael Press === Subject: Application of Birthday Paradox Hi can I use the paradox for the following problem? I have N bins and M balls. Balls are thrown randomly over the N bins. Each bin can have more than 1 ball. What is the chance of choosing a bin with at least 1 ball in it? If you are giving k consecutive bins to choose, what is the probability? === Subject: Re: Simple questions about Galois fields Since others have answered some of your questions, let me just reply to a few points. > Morning all, > Here's what i don't get: What do you call the polynomial that you use as a modulus when multiplying > elements? I've seen one source call it a primitive polynomial, one a prime > polynomial, and one a defining polynomial. Many texts manage to discuss it > without ever giving it a name! Defining polynomial would be OK. Primitive polynomial means something stronger. A primitive polynomial is an irreducible polynomial of degree n over the ground field with the property that the roots are generators of the (cyclic) multiplicative group of the extension field that they define. So for a polynomial of degree n over GF(p), the roots of a primitive polynomial would have multiplicative order p^n - 1 in GF(p^n). You may wish to use a primitive polynomial as defining polynomial, but this is not essential. For example, for p=2 and n=4, x^4+x+1 is primitive (roots have order 15 in GF(16)), but x^4+x^3+x^2+x+1 is not (roots have order 5). You could however use either of these as defining polynomial. I am not sure what prime polynomial means. > I understand why this polynomial has to be order n+1 and monic, and i have > a vague idea why it has to be irreducible (same reason p has to be prime > for prime fields). But Wikipedia says it should be a 'minimal polynomial' > of a generator of the field - and that a minimal polynomial of some x is > the smallest one which, when evaluated on x, is zero. This has got me > stumped. For starters, the x you're evaluating it on is an element of the > field, and so is itself a polynomial - so you substitute the little > polynomial into the x's in the big one, and then multiply everything out? > And that somehow comes to zero? What? Why? Is the polynomial x (ie 1x + 0) always a multiplicative generator of a > Galois field? If not, how do you find a generator? As explained above, it is a generator if and only if the defining polynomial is primitive. If not, then you will usually find a generator quickly by considering random elements. > What's this Conway polynomial business? Firstly, am i right in thinking > it's nothing to do with the Conway polynomials in knot theory? I know > nothing about knot theory (except that i now have a book on 85 ways to tie > a tie because of it, and that some loon has worked out what the Gordian > knot was), but that's what Wikipedia wants to tell me about. Secondly, how > does the lexicographical ordering work? Lexicographical ordering in > general, fine, but there's some sort of stuff about alternating signs on > the coefficients that escapes me completely. What's happening here, and > why? What's the compatibility condition about? And after all that, why are > Conway polynomials important? You might have gathered that one of the difficulties with this topic is that there is no natural way of choosing a defining polynomial of degree n over GF(p) which could be agreed upon as a standard. The Conway polynomial is an attempt to define such a standard. They are generally used as the default by computer algebra systems for constructing finite fields. Conway polynomials are always primitive, and they are chosen to be compatible with each other in the sense that the roots of the Conway polynomial of degree n are powers of the roots of degree m for all multiples m of n. They are then defined as the least polynomials that satisfy those conditions under a specific ordering of the set of all polynomials over GF(p). This order, which involves a lexicographical ordering of the coefficients is necessarily somewhat arbitrary, but this is a consequence of the lack of a natural choice of a canonical polynomial. One of the practical problems with Conway polynomials is that they are not easy to compute when n and p get large - in fact the difficulty of computing them increases unpleasantly with n. Last time I checked they had been computed up to 2^127, 3^61, 5^31, 7^31, .., 127^6, .. Of course they only need to be computed once! Derek Holt. === Subject: Re: Simple questions about Galois fields > Morning all, for any answers you may have. I got a new ADSL service connected recently, so i read up on how ADSL > works; that led me to Reed-Solomon codes, and that led me to Galois > fields. Being a sucker for discrete mathematics, i'm trying to understand > them. Not being a mathematician (i'm a biologist by trade, and secretly a > computer programmer), this is slow going. Specifically, it's the polynomials. Prime fields, i get, no problem - > GF(p) is Z_p under modular addition and multiplication; easy. I now > understand the structure of extension fields - GF(p^n) is the set of > order-n polynomials over GF(p) - and how you do addition. Having got my > head round how you do division with polynomials, i now more or less > understand how you do multiplication too. Here's what i don't get: What do you call the polynomial that you use as a modulus when multiplying > elements? I've seen one source call it a primitive polynomial, one a prime > polynomial, and one a defining polynomial. Many texts manage to discuss it > without ever giving it a name! I understand why this polynomial has to be order n+1 and monic, and i have > a vague idea why it has to be irreducible (same reason p has to be prime > for prime fields). But Wikipedia says it should be a 'minimal polynomial' > of a generator of the field - and that a minimal polynomial of some x is > the smallest one which, when evaluated on x, is zero. This has got me > stumped. For starters, the x you're evaluating it on is an element of the > field, and so is itself a polynomial - so you substitute the little > polynomial into the x's in the big one, and then multiply everything out? > And that somehow comes to zero? What? Why? Is the polynomial x (ie 1x + 0) always a multiplicative generator of a > Galois field? If not, how do you find a generator? What's this Conway polynomial business? Firstly, am i right in thinking > it's nothing to do with the Conway polynomials in knot theory? I know > nothing about knot theory (except that i now have a book on 85 ways to tie > a tie because of it, and that some loon has worked out what the Gordian > knot was), but that's what Wikipedia wants to tell me about. Secondly, how > does the lexicographical ordering work? Lexicographical ordering in > general, fine, but there's some sort of stuff about alternating signs on > the coefficients that escapes me completely. What's happening here, and > why? What's the compatibility condition about? And after all that, why are > Conway polynomials important? How come all Galois fields of a given order are isomorphic? Surely, if > they have different defining polynomials, they'll do multiplication > differently, and since you've also got the additive structure in there, > you can't just relabel all the elements to make them look the same. Does > the multiplicative structure come out the same regardless of the defining > polynomial, or is there always a way to work the relabelling? I have to say, dialup was never this complicated. Oh, bonus question: what does the notation Z/3Z, where the Zs are > double-struck, ie meaning the integers, mean? Is it the same as > Z-underscore-3? tom -- > Tomorrow has made a phone call to today. Before I try to answer this, let me just recommend that, if you have an easily accessible library, you borrow a textbook on algebra. There's tons of them out there that all cover the basics of Galois Theory. Here's a few at the undergrad level (titles might be a little off, authors are correct): I.N. Herstein, Topics in Algebra Dummit & Foote, Modern Algebra W. Keith Nicholson, An Introduction to Abstract Algebra I believe Rotman and Lang also have undergrad algebra texts that I'm not familiar with; just be sure to not mix them up with their graduate versions. I think your biggest confusion right now is what the elements of GF(p^n) consist of. They are not degree n polynomials over GF(p) (which isn't even a field). Ignoring the question of uniqueness for the moment, say that f is an irreducible polynomial of degree n over GF(p), and let c be any root. Then GF(p^n) is the field obtained by adjoining c to GF(p). What this means is that GF(p^n) is the smallest field extension of GF(p) containing c. As can be proved, GF(p^n) is a vector space of dimension n over GF(p) with basis given by {1, c, c^2, ..., c^(n-1)}. The polynomial f is called the minimal polynomial for c, the defining polynomial for c, or the irreducible polynomial of c over GF(p). They all mean the same thing. It can also be proved that the GF(p^n) obtained above is independent of the irreducible polynomial chosen as well as the root adjoined, in the sense that they are all isomorphic. There are probably a few ways of proving this, but the slickest is to show that GF(p^n) is the so-called splitting field of the polynomial x^(p^n) - x over GF(p), and to then use the fact that any two splitting fields of the same polynomial are isomorphic. If you're curious, I really do recommend reading about this in a textbook if possible. At this point, you may feel really baffled about what these fields look like, and how elements are multiplied. To get a feel for it, you might want to try this example, or something like it. The polynomial x^3 - x + 1 is irreducible over GF(3). This is easy to see because it has no roots in GF(3). Then GF(27) is obtained by adjoining any root of this polynomial. So if c is a root, GF(27) has {1, c, c^2} as a basis over GF(p). Try to see if you can write out what all 27 elements are in terms of this basis, and then play around with them to see how they multiply (remember, all you know about c is that c^3 - c + 1 = 0). As for your bonus question, Z/3Z is the integers mod 3, often written Z_3. If you want to learn more about his, Z is an example of something called a ring, 3Z is an ideal of this ring, and Z/3Z is the quotient ring. Again, this would be covered in any basic algebra text. Hopefully this made some sense. If not, I'm sure somebody else can explain things better. Mike === Subject: Re: Simple questions about Galois fields > Morning all, > for any answers you may have. > I got a new ADSL service connected recently, so i read up on how ADSL > works; that led me to Reed-Solomon codes, and that led me to Galois > fields. Being a sucker for discrete mathematics, i'm trying to understand > them. Not being a mathematician (i'm a biologist by trade, and secretly a > computer programmer), this is slow going. > Specifically, it's the polynomials. Prime fields, i get, no problem - > GF(p) is Z_p under modular addition and multiplication; easy. I now > understand the structure of extension fields - GF(p^n) is the set of > order-n polynomials over GF(p) - and how you do addition. Having got my > head round how you do division with polynomials, i now more or less > understand how you do multiplication too. > Here's what i don't get: > What do you call the polynomial that you use as a modulus when multiplying > elements? I've seen one source call it a primitive polynomial, one a prime > polynomial, and one a defining polynomial. Many texts manage to discuss it > without ever giving it a name! > I understand why this polynomial has to be order n+1 and monic, and i have > a vague idea why it has to be irreducible (same reason p has to be prime > for prime fields). But Wikipedia says it should be a 'minimal polynomial' > of a generator of the field - and that a minimal polynomial of some x is > the smallest one which, when evaluated on x, is zero. This has got me > stumped. For starters, the x you're evaluating it on is an element of the > field, and so is itself a polynomial - so you substitute the little > polynomial into the x's in the big one, and then multiply everything out? > And that somehow comes to zero? What? Why? > Is the polynomial x (ie 1x + 0) always a multiplicative generator of a > Galois field? If not, how do you find a generator? > What's this Conway polynomial business? Firstly, am i right in thinking > it's nothing to do with the Conway polynomials in knot theory? I know > nothing about knot theory (except that i now have a book on 85 ways to tie > a tie because of it, and that some loon has worked out what the Gordian > knot was), but that's what Wikipedia wants to tell me about. Secondly, how > does the lexicographical ordering work? Lexicographical ordering in > general, fine, but there's some sort of stuff about alternating signs on > the coefficients that escapes me completely. What's happening here, and > why? What's the compatibility condition about? And after all that, why are > Conway polynomials important? > How come all Galois fields of a given order are isomorphic? Surely, if > they have different defining polynomials, they'll do multiplication > differently, and since you've also got the additive structure in there, > you can't just relabel all the elements to make them look the same. Does > the multiplicative structure come out the same regardless of the defining > polynomial, or is there always a way to work the relabelling? > I have to say, dialup was never this complicated. > Oh, bonus question: what does the notation Z/3Z, where the Zs are > double-struck, ie meaning the integers, mean? Is it the same as > Z-underscore-3? > tom > -- > Tomorrow has made a phone call to today. Before I try to answer this, let me just recommend that, if you have an > easily accessible library, you borrow a textbook on algebra. There's > tons of them out there that all cover the basics of Galois Theory. > Here's a few at the undergrad level (titles might be a little off, > authors are correct): I.N. Herstein, Topics in Algebra > Dummit & Foote, Modern Algebra > W. Keith Nicholson, An Introduction to Abstract Algebra I believe Rotman and Lang also have undergrad algebra texts that I'm > not familiar with; just be sure to not mix them up with their graduate > versions. I think your biggest confusion right now is what the elements of > GF(p^n) consist of. They are not degree n polynomials over GF(p) > (which isn't even a field). Ignoring the question of uniqueness for > the moment, say that f is an irreducible polynomial of degree n over > GF(p), and let c be any root. Then GF(p^n) is the field obtained by > adjoining c to GF(p). What this means is that GF(p^n) is the smallest > field extension of GF(p) containing c. As can be proved, GF(p^n) is a > vector space of dimension n over GF(p) with basis given by {1, c, c^2, > ..., c^(n-1)}. The polynomial f is called the minimal polynomial for > c, the defining polynomial for c, or the irreducible polynomial of c > over GF(p). They all mean the same thing. It can also be proved that the GF(p^n) obtained above is independent of > the irreducible polynomial chosen as well as the root adjoined, in the > sense that they are all isomorphic. There are probably a few ways of > proving this, but the slickest is to show that GF(p^n) is the so-called > splitting field of the polynomial x^(p^n) - x over GF(p), and to then > use the fact that any two splitting fields of the same polynomial are > isomorphic. If you're curious, I really do recommend reading about > this in a textbook if possible. At this point, you may feel really baffled about what these fields > look like, and how elements are multiplied. To get a feel for it, > you might want to try this example, or something like it. The > polynomial x^3 - x + 1 is irreducible over GF(3). This is easy to see > because it has no roots in GF(3). Then GF(27) is obtained by adjoining > any root of this polynomial. So if c is a root, GF(27) has {1, c, c^2} > as a basis over GF(p). Try to see if you can write out what all 27 > elements are in terms of this basis, and then play around with them to > see how they multiply (remember, all you know about c is that c^3 - c + > 1 = 0). As for your bonus question, Z/3Z is the integers mod 3, often written > Z_3. If you want to learn more about his, Z is an example of something > called a ring, 3Z is an ideal of this ring, and Z/3Z is the quotient > ring. Again, this would be covered in any basic algebra text. Hopefully this made some sense. If not, I'm sure somebody else can > explain things better. Mike I just read Gerry's post above and realized what you meant by saying that GF(p^n) consisted of degree n polynomials over GF(p). As he explained, it is, modulo an irreducible polynomial of degree n. Sorry if i caused any confusion. Mike === Subject: Re: Simple questions about Galois fields > What do you call the polynomial that you use as a modulus when multiplying > elements? I've seen one source call it a primitive polynomial, one a prime > polynomial, and one a defining polynomial. Many texts manage to discuss it > without ever giving it a name! >I understand why this polynomial has to be order n+1 and monic, and i have > a vague idea why it has to be irreducible (same reason p has to be prime > for prime fields). But Wikipedia says it should be a 'minimal polynomial' > of a generator of the field - and that a minimal polynomial of some x is > the smallest one which, when evaluated on x, is zero. This has got me > stumped. For starters, the x you're evaluating it on is an element of the > field, and so is itself a polynomial - so you substitute the little > polynomial into the x's in the big one, and then multiply everything out? > And that somehow comes to zero? What? Why? >Is the polynomial x (ie 1x + 0) always a multiplicative generator of a > Galois field? If not, how do you find a generator? Maybe best to look at an example. Field of 4 elements. We need a polynomial of degree 2, irreducible over Z_2; f = x^2 + x + 1 will do. The highbrow way of describing GF(4) is, you take Z_2[x], the ring of polynomials with coefficients in Z_2, and mod out by the ideal generated by f, so GF(4) = Z_2[x] / (f). The elements of GF(4) are then the cosets of (f), which can be taken to be 0 + (f), 1 + (f), x + (f), and x + 1 + (f). But we usually go lowbrow, and just take GF(4) to be the set of coset representatives, that is, GF(4) = {0, 1, x, x + 1}, that is, the polynomials of degree less than 2. Now I really don't care what you call f - defining polynomial is as good a term as any, I suppose. So, all calculation in GF(4) is done modulo 2 and modulo f. Now, forgetting that x is a polynmomial, and just thinking of it as an element of a field - practically a number - what is its minimal polynomial? Well, f(x) is zero in GF(4), so f is its minimal polynomial. Now let's go a bit more general. Take Z_p, take some polynomial of degree n + 1 and irreducible over Z_p, then GF(p^n) is Z_p[x] / (f), which you can think of as the polynomials of degree at most n with coefficients in Z_p, with arithmetic done mod p and mod f. It can be proved that the nonzero elements of GF(p^n) form a cyclic group under multiplication - that is, there is an element y such that every nonzero element of GF(p^n) is a power of y. It may be that x is such an element - it was, in our GF(4) example - or it may not be. Such an element is called primitive, and its minimal polynomial is called a primitive polynomial. You don't need to use a primitive polynomial f when you construct GF(p^n), but there are some advantages to using one. The disadvantage is that first you have to find one. There's no really really simple way to do that. Put another way, there's no really, really simple way to find a multiplicative generator - that's already true when n = 1 (quick: what's a multipplicative generator of Z_p when p = 55079?). > What's this Conway polynomial business? I don't know. > How come all Galois fields of a given order are isomorphic? Surely, if > they have different defining polynomials, they'll do multiplication > differently, and since you've also got the additive structure in there, > you can't just relabel all the elements to make them look the same. Does > the multiplicative structure come out the same regardless of the defining > polynomial, or is there always a way to work the relabelling? Why don't you try some examples? For GF(9), say, find a bunch of different irreducible quadratics over Z_3, use each of them to construct GF(9), write out the group tables, and see for yourself how the relabelling goes. > Oh, bonus question: what does the notation Z/3Z, where the Zs are > double-struck, ie meaning the integers, mean? Is it the same as > Z-underscore-3? That depends on what you mean by Z_3. Again, there's a low road and a high road. The low road is Z_3 is the set {0, 1, 2} with arithmetic modulo 3. The high road is Z_3 is Z / 3 Z, which is the set of cosets of the ideal 3 Z = {..., -6, -3, 0, 3, 6, ...} in the ring Z, with arithmetic on the cosets being done by means of coset representatives. But if we write 0, 1, and 2, respectively, for the cosets 0 + 3 Z, 1 + 3 Z, and 2 + 3 Z, we see there's not much real difference between the two ways of doing things. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Simple questions about Galois fields >> Oh, bonus question: what does the notation Z/3Z, where the Zs are >> double-struck, ie meaning the integers, mean? Is it the same as >> Z-underscore-3? >That depends on what you mean by Z_3. Again, there's a low road and > a high road. The low road is Z_3 is the set {0, 1, 2} with arithmetic > modulo 3. The high road is Z_3 is Z / 3 Z, which is the set of cosets > of the ideal 3 Z = {..., -6, -3, 0, 3, 6, ...} in the ring Z, with > arithmetic on the cosets being done by means of coset representatives. > But if we write 0, 1, and 2, respectively, for the cosets 0 + 3 Z, > 1 + 3 Z, and 2 + 3 Z, we see there's not much real difference between > the two ways of doing things. I would have thought Z_3 was used rather more often today for the ring of 3-adic integers. Not meant as criticism of your excellent tutorial. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Simple questions about Galois fields days. My association with the Department is that of an alumnus. >I would have thought Z_3 was used rather more often today >for the ring of 3-adic integers. Depends entirely on who you talk to. For a number theorist, Z_p ->definitely<- represents the ring of p-adic integers. They seem to invariably prefer Z/nZ for the quotient ring of integers modulo n. (I took abstract algebra from a number theorist, and he objected strongly to using Z_n for the integers modulo n). Ring theorists, on the other hand, who use the quotient far more often to the p-adics, use the simpler notation Z_n for integers modulo n. (Group theorists seem to be somewhat divided; I've gotten referees quite annoyed at me for using Z/nZ for the additive cyclic group of order n, and insist I use something like C_n instead, saying that Z/nZ is reserved for the ->ring<-; and other referees tell me to drop the ad hoc notation C_n and stick to the standard Z/nZ. So you never know). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Simple questions about Galois fields >> Oh, bonus question: what does the notation Z/3Z, where the Zs are > double-struck, ie meaning the integers, mean? Is it the same as > Z-underscore-3? >> >> That depends on what you mean by Z_3. Again, there's a low road and >> a high road. The low road is Z_3 is the set {0, 1, 2} with arithmetic >> modulo 3. The high road is Z_3 is Z / 3 Z, which is the set of cosets >> of the ideal 3 Z = {..., -6, -3, 0, 3, 6, ...} in the ring Z, with >> arithmetic on the cosets being done by means of coset representatives. >> But if we write 0, 1, and 2, respectively, for the cosets 0 + 3 Z, >> 1 + 3 Z, and 2 + 3 Z, we see there's not much real difference between >> the two ways of doing things. I would have thought Z_3 was used rather more often today >for the ring of 3-adic integers. The difference may not be merely one of today versus some other time (presumably yesterday, or yesteryear); at least in the early 1970s, topologists very much favored Z/3Z for the quotient ring and Z_3 for the 3-adic integers (and Z_(3) for Z localized at 3, to boot). I don't hang out with that kind of topologist any more so don't know what they do today, but suspect they do now what they did then. Lee Rudolph === Subject: Is this true ? ( My apologize ) Hello everybody Yesterday I answered about this : If f (x) >= 0 then f ( x/n ) * f ( 1 - x/n ) >= [ f ( x/n ) ] ^ n for each n positve integer. was true or false. I commeted a mistake. The inequation is f ( x/n ) * f [ (1 - 1/n)x ] >= [ f ( x/n ) ] ^ n Now, again, is this true ? === Subject: Is this true ? ( My apologize ) Hello everybody Yesterday I answered about this : If f (x) >= 0 then f ( x/n ) * f ( 1 - x/n ) >= [ f ( x/n ) ] ^ n for each n positve integer. was true or false. I commeted a mistake. The inequation is f ( x/n ) * f [ (1 - 1/n)x ] >= [ f ( x/n ) ] ^ n Now, again, is this true ? === Subject: Re: Is this true ? ( My apologize ) > If f (x) >= 0 then f ( x/n ) * f ( 1 - x/n ) >= [ f ( > x/n ) ] ^ n for each n positve integer. was true or false. I commeted a mistake. The inequation is f ( x/n ) * f [ (1 - 1/n)x ] >= [ f ( x/n ) ] ^ n Now, again, is this true ? > No, same counter example f(x) = x^2, x = 1, n = 1. === Subject: Since the beginning.... Hello IÇll try to explain the question Is this true ? . I am studying Continuous Time Markov Chains Theory. In this scenario, we consider a special probability functions since the state i to the state j , P(i,j) ( t ) with three properties : ( 1 ) P(i,j) ( t ) >= 0 for all i, j in E ( E = set of states ) and for all t>=0 and Sum ( P(i,j) ( t ) ) = 1 ( Sumarize in j in E ) for all i in E and for all t >=0 ( 2 ) P(i,j) ( s + t ) = Sum ( P(i,k)(s) * P(k,j)(t) ) ( Sumarize in k in E ) for all i, j in E and for all s,t >=0 ( 3 ) P(i,j)(0) = 0 if i <> j and P(i,j)(0) = 1 if i = j There is a Theorem in my book and I donÇt understand a step in the proof. The Theorem is : For all i in E and for all t >=0 , P(i,i) > 0 P(i,i) = Sum ( P(i,k)( t/n ) * P(k,i)( (1-1/n)*t ) ) >= P(i,i)( t/n ) * P(i,i)( (1-1/n)*t ) with independence the value of n for all n integer. So ( and this is MY DOUBT ) : P(i,i)(t) >= [ P(i,i)( t/n ) ]^n for all n integer. After this, if you fix t > 0, P(i,i) ( t/n ) > 0 because P(i,i)(h) ---> 1 if h ---> 0 ( we supose continuity in 0 ) and we conclude that P(i,i)( t ) > 0 This is the real question. === Subject: Re: ln(2)/2 proof Hello Mr Ullrich >> Rob suggests grouping the terms in the series in groups of three. >> Regarding why that would prove what's needed, you should note this: >> If you have a series sum a_n, if you define a new series sum b_n >> by saying that b_n is the sum of a group of consecutive a_j's >> (in the same order), then it follows that sum a_n converges >> if (i) sum b_n converges, (ii) a_n -> 0, and (iii) the >> number of terms in the groups defining the b_n is _bounded_. >> (You need to show that sum_n^m a_j -> 0 as n,m -> infinity. >> But sum_n^m a_j is equal to sum_N^M b_k plus boundedly >> many a_j's...) I think more about this and would like to try again. I am to study sum(n=0,oo) (-1)^n / (z+n) z is complex number. Since what i learn i know i can write this sum(n=0,oo) 1/ (z+2n) - 1/ (z+2n+1) = sum(n=0,oo) 1/ ((z+2n)(z+2n+1)) since i no change order of terms and each sub series has finite many >terms. But first look to fact of z = 0,-1, -2 ,-3, -4...-n then >term has singularity so i say series then diverge by this singularity. I am then to look at series for z not equal to negative integers and 0. > Before i continue is what i have done correct. May i write series as >above and may i say it diverge since term has singularity. If z is not a non-positive integer then yes, what you did shows that the series converges. In fact if you define f(z) to be that sum then you can use the argument to show the series converges uniformly on compact subsets of S = C {0,-1,-,2,...}, so that the sum is holomorphic in S. If z is a negative integer I don't think it's quite correct to say that the series diverges - for a series to diverge the series has to _exist_, and if z = -n then one of the terms in the series is simply undefined. It _is_ true that f(z) has a pole at -n. (probably the best way to show this is to write f(z) as 1/(z+n) plus the sum of the other terms; the term 1/(x+n) has a pole at -n, and the sum of the other terms converges uniformly in a neighborhood of -n. ************************ David C. Ullrich === Subject: Re: Spec(C[X,Y]/(XY-1)] <45296CD0.5000706@web.de> <452A324B.10303@web.de> <45332E9C.8010600@web.de> <4533B6F8.8090604@web.de> svuytdehaage@hotmail schreef: > Jannick Asmus schreef: > Jannick Asmus schreef: > Jannick Asmus schreef: >>something else than C* in that case? >> What primes does Spec(C[X,Y]/(XY-1)) contain besides maximal ideals? >> Where should it map to? >When you write en element of the ring as the sum over i of a_i X^i and >everything divided by x^j for some j, then the maximal ideals are (X-c) >cneq 0. X-0 = X is also not a prime ideal, so I think the primes and >maximal ideals coincide here. >> The quotient ring is an integral domain, isn't it? Hence the zero-ideal >> therein is prime, but not maximal. ;) > J. > Hmmm, yes... But then I don't understand why it is (not, right?) C*... > Well, we have maximal ideals corresponding a unique complex non-zero > figure and, in addition, the prime ideal in R := C[X,Y]/(XY-1) which is > not maximal. Where does it map to if it is considered as an element of > Spec(R)? I think by writing Spec(R) you mean the set of maximal prime > ideals only, don't you? > J. Indeed, I was wrong. We're only considering max. ideals, but normally I've another question in the same subject. I hope you still want to give me a hint. I want all maximal ( ) ideals of C[X,Y,X(Y)^(-1), Y(X)^(-1)], i.e. the ring with monomials in X and Y, with total degree always non-negative. I thought that this ring is iso to C[X,Y,X^(-1),Y^(-1)], so the maximals are iso to C* x C*. Is this === Subject: Re: Spec(C[X,Y]/(XY-1)] <45296CD0.5000706@web.de> <452A324B.10303@web.de> <45332E9C.8010600@web.de> <4533B6F8.8090604@web.de> svuytdehaage@hotmail schreef: > svuytdehaage@hotmail schreef: > Jannick Asmus schreef: >Jannick Asmus schreef: >Jannick Asmus schreef: >>something else than C* in that case? >> What primes does Spec(C[X,Y]/(XY-1)) contain besides maximal ideals? >> Where should it map to? >When you write en element of the ring as the sum over i of a_i X^i and >everything divided by x^j for some j, then the maximal ideals are (X-c) >cneq 0. X-0 = X is also not a prime ideal, so I think the primes and >maximal ideals coincide here. >> The quotient ring is an integral domain, isn't it? Hence the zero-ideal >> therein is prime, but not maximal. ;) > J. >Hmmm, yes... But then I don't understand why it is (not, right?) C*... > Well, we have maximal ideals corresponding a unique complex non-zero >figure and, in addition, the prime ideal in R := C[X,Y]/(XY-1) which is >not maximal. Where does it map to if it is considered as an element of >Spec(R)? I think by writing Spec(R) you mean the set of maximal prime >ideals only, don't you? > J. > Indeed, I was wrong. We're only considering max. ideals, but normally I've another question in the same subject. I hope you still want to > give me a hint. I want all maximal ( ) ideals of C[X,Y,X(Y)^(-1), > Y(X)^(-1)], i.e. the ring with monomials in X and Y, with total degree > always non-negative. I thought that this ring is iso to > C[X,Y,X^(-1),Y^(-1)], so the maximals are iso to C* x C*. Is this Edit; I made a mistake above, it is of course not iso to C[X,Y,X^(-1),Y^(-1)], but how can I then find the max. ideals? === Subject: Re: Spec(C[X,Y]/(XY-1)] > svuytdehaage@hotmail schreef: > >> svuytdehaage@hotmail schreef: > Jannick Asmus schreef: >> Jannick Asmus schreef: >> Jannick Asmus schreef: >> something else than C* in that case? >> What primes does Spec(C[X,Y]/(XY-1)) contain besides maximal ideals? >> Where should it map to? > When you write en element of the ring as the sum over i of a_i X^i and > everything divided by x^j for some j, then the maximal ideals are (X-c) > cneq 0. X-0 = X is also not a prime ideal, so I think the primes and > maximal ideals coincide here. >> The quotient ring is an integral domain, isn't it? Hence the zero-ideal >> therein is prime, but not maximal. ;) >> J. > Hmmm, yes... But then I don't understand why it is (not, right?) C*... >> Well, we have maximal ideals corresponding a unique complex non-zero >> figure and, in addition, the prime ideal in R := C[X,Y]/(XY-1) which is >> not maximal. Where does it map to if it is considered as an element of >> Spec(R)? I think by writing Spec(R) you mean the set of maximal prime >> ideals only, don't you? >> J. > Indeed, I was wrong. We're only considering max. ideals, but normally >> I've another question in the same subject. I hope you still want to >> give me a hint. I want all maximal ( ) ideals of C[X,Y,X(Y)^(-1), >> Y(X)^(-1)], i.e. the ring with monomials in X and Y, with total degree >> always non-negative. I thought that this ring is iso to >> C[X,Y,X^(-1),Y^(-1)], so the maximals are iso to C* x C*. Is this >Edit; I made a mistake above, it is of course not iso to > C[X,Y,X^(-1),Y^(-1)], but how can I then find the max. ideals? Well, let's focus on maximal ideals only. ;) Consider the inclusion map i: C[X,Y] -> R:=C[X,Y,X/Y,Y/X] which is a homomorphism of finitely generated C-algebras. If M is a maximal ideal of R, the prime ideal M':=i^-1(M) is maximal in C[X,Y] and the induced C-algebra homomorphisms C -> C[X,Y]/M' -> R/M are both isomorphisms, by Hilbert's Nullstellensatz. Now M' is generated by X-a and Y-b with (a,b) in C^2. Then ... Can you carry on from here? HTH. J. === Subject: Qoutient fields of factor rings of an integral domain Ok, I know my question is silly now.. but Im beginning to doubt my knowledge here so I would like some advises here. Is it true that given an integral domain R and any prime ideal p of it, then we can regard Qout(R/p) as a subring of Qout(R)? .. I can see this for all the integral domains Im working with, but I dont know if Im just imagining it :) .. ... uh.. wait.. Qout(Z/(2)) can never be regarded as as subring of Qout(Z)... ok .. I guess this post can be disregarded :) Jose Capco === Subject: Re: simplifying trig expression >> I'm working on some homework and came up with the right answer, but the >> book goes further into simplifying this expression, but I don't >> understand where it comes from. First thought was a double-angle >> formula. Unfortunately, it has been a long time since pre-calculus :) >> 2y * cos(t) - x * sin(t) >> 2[cos(t)^2 - sin(t)^2] >> How is the second expression derived from the first? >I don't know, unless y = cos(t) and x = 2 sin(t), in which case > it's just substitution. >State the problem in its entirety. > --- Christopher Heckman > This is a problem that involves using the chain rule for multiple variables. You guessed right, y=cos(t) and x= 2*sin(t), but how do they get back in the equation? The complete problem and break-down of steps is here: http://img87.imageshack.us/img87/7518/se12e01005um8.gif. The part I'm having problems with is problem (a), going from step 2 to step 3. I'm fine with everything else. -- Mike === Subject: Re: simplifying trig expression > I'm working on some homework and came up with the right answer, but the > book goes further into simplifying this expression, but I don't > understand where it comes from. First thought was a double-angle > formula. Unfortunately, it has been a long time since pre-calculus :) >> 2y * cos(t) - x * sin(t) > 2[cos(t)^2 - sin(t)^2] >> How is the second expression derived from the first? >> >> I don't know, unless y = cos(t) and x = 2 sin(t), in which case >> it's just substitution. >> >> State the problem in its entirety. >> >> --- Christopher Heckman >> >This is a problem that involves using the chain rule for multiple >variables. You guessed right, y=cos(t) and x= 2*sin(t), but how do they >get back in the equation? The complete problem and break-down of steps is here: >http://img87.imageshack.us/img87/7518/se12e01005um8.gif. The part I'm >having problems with is problem (a), going from step 2 to step 3. I'm >fine with everything else. > You would do just what the post from Christopher Heckman (Proginoskes) indicated. Given y = cos(t) x = 2 sin(t) (2) = 2 y cos(t) - x sin(t) Substituting for y and x, = 2 cos(t) cos(t) - 2 sin(t) sin(t) = 2 cos^2(t) - 2 sin^2(t) (3) = 2 ( cos^2(t) - sin^2(t) ) HTH === Subject: Re: simplifying trig expression >> I'm working on some homework and came up with the > right answer, but the >> book goes further into simplifying this > expression, but I don't >> understand where it comes from. First thought was > a double-angle >> formula. Unfortunately, it has been a long time > since pre-calculus :) >> 2y * cos(t) - x * sin(t) >> 2[cos(t)^2 - sin(t)^2] >> How is the second expression derived from the > first? >I don't know, unless y = cos(t) and x = 2 > sin(t), in which case > it's just substitution. >State the problem in its entirety. > --- Christopher Heckman >This is a problem that involves using the chain rule > for multiple > variables. You guessed right, y=cos(t) and x= > 2*sin(t), but how do they > get back in the equation? >The complete problem and break-down of steps is here: >http://img87.imageshack.us/img87/7518/se12e01005um8.gi > f. The part I'm > having problems with is problem (a), going from step > 2 to step 3. I'm > fine with everything else. -- > Mike Exactly as Proginoskes said: substitution. You are told that dy/dx= 2y cos t- x sin t and that x= 2 sin t, y= cos t. Replacing x and y by those, dy/dt= 2(cos t)(cos t)- (2 sin t)(sin t) dy/dt= 2 cos^2 t- 2 sin^2 t dy/dt= 2(cos^2 t- sin^2 t). === Subject: Re: simplifying trig expression > I'm working on some homework and came up with the right answer, but the > book goes further into simplifying this expression, but I don't > understand where it comes from. First thought was a double-angle > formula. Unfortunately, it has been a long time since pre-calculus :) >> 2y * cos(t) - x * sin(t) > 2[cos(t)^2 - sin(t)^2] >> How is the second expression derived from the first? >> I don't know, unless y = cos(t) and x = 2 sin(t), in which case >> it's just substitution. >> State the problem in its entirety. >> --- Christopher Heckman > This is a problem that involves using the chain rule for multiple > variables. You guessed right, y=cos(t) and x= 2*sin(t), but how do they > get back in the equation? >The complete problem and break-down of steps is here: > http://img87.imageshack.us/img87/7518/se12e01005um8.gif. The part I'm > having problems with is problem (a), going from step 2 to step 3. I'm > fine with everything else. Seems like I answered my own question :) Turns out you do have to substitute the x and y equations back in after applying the chain rule -- Mike === Subject: Re: Elementary Group Theory days. My association with the Department is that of an alumnus. >** This is a homework question ** Exercise: Suppose that G is a group with more than one >element and G has no proper, non-trivial subgroups. >Prove that |G| is prime. (Do not assume at the outset >that G is finite.) Solution: For any non-trivial g in G we must have = G >since is a subgroup and G has no proper, non-trivial >subgroups. If = {e, g}, then we are done since this >implies |G| = 2. If not then choose any non-trivial >g^k in . It follows that = G = and so >we have |g^k| = |G| = |g|. Now recall that |g^k| = |g| / gcd{|g|, k}. You are here assuming that G is finite. Otherwise, |g| is infinite (or 0, depending on your definition) and this does not make sense. So you must FIRST establish that G is nor infinite. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Elementary Group Theory > ** This is a homework question ** >Exercise: Suppose that G is a group with more than one > element and G has no proper, non-trivial subgroups. > Prove that |G| is prime. (Do not assume at the outset > that G is finite.) >Solution: For any non-trivial g in G we must have = G > since is a subgroup and G has no proper, non-trivial > subgroups. If = {e, g}, then we are done since this > implies |G| = 2. If not then choose any non-trivial > g^k in . It follows that = G = and so > we have |g^k| = |G| = |g|. Now recall that >|g^k| = |g| / gcd{|g|, k}. >Thus, gcd{|G|, k} = 1 for any integer k which implies > |G| itself is prime. [] >I'm a bit sketchy about my proof because the exercise > occurs in the chapter about cosets and Lagrange's theorem > and I made no direct use of these. >I was also thinking maybe I should just consider the > left or right coset decompositions of G. By hypothesis > these would either be G = G or G = {e} / {g_1} / ..., > which seems to me like it should force |G| to be prime, > but I just couldn't formalize the argument in this > manner. >Any comments, hints, or suggestions? >TIA, > Kyle Czarnecki Alternatively, you could consider the surjective group homomorphism Z -> G mapping 1 to g =/= e. Then consider the quotients of Z ... This approach does not use Lagrange's theorem explicitly, but implicitly it can be recognized. HTH. J. === Subject: Re: Cartesian product non-associativity days. My association with the Department is that of an alumnus. >> Regularity states that for every nonempty set A, there exists x in A >> such that x/A is empty. >> Take the Kuratowski definition of ordered pair, (a,b)={{a},{a,b}}. >> Let A be a nonempty set, and let >> C = A / { x in P(A) : x is a singleton} >> If A = A x B, then for every a in A there exists a' in A such that >> {a'} is in a (since there exists a' in A and b in B such that >> a = (a',b) = {{a'},{a',b}}, so {a'} is in (a',b)=a). >> Now consider C. According to regularity, there exists x in C such that >> x/C is empty. >> If x is in A, then there exists a' in A such that {a'} is in >> x. But {a'} is in {y in P(A) : y is a singleton}, so {a'} is in both x >> and C. Thus, x is not in A. >> Therefore, x is of the form {a} for some a in A. But then a is an >> element of both C and x, again contradicting that x is disjoint from >> C. >If you don't mind, would you say something about the train of thought >that led you to that proof? What gave you the idea for that >construction? No problem. Though it may make me look less impressive. (-; I figured from your comments that some proof without Choice would be possible. But the infinite descending chain argument would almost certainly not work without some form of Choice. So I looked up a few equivalents of Regularity, and found the one I quoted. That looked like the way to go. Of course, it wasn't ->quite<- right; for any a in A we could show an a' in A with {a'} in a, so we would get a chain a' in {a'} in a. First I tried to show it directly: let a in A be an element disjoint from A. Okay, I know there is {a'} in a with a in A... but how does this give a contradiction? It seemed like it wasn't going to prosper easily. So then I tried to modify A so that we could strip away one layer of brackets in some way, but there was no obvious way to do it. That's when it hit me that I should try going the other way: just ->add<- the element {a'} to A. The obvious thing to worry about is whether the element you add happens to be disjoint to A, but this doesn't happen here, obviously. Ah, but what about a', then. Well, just add {a''} to A... No, I'm just going down dependent choice again... Well, just throw in ->all<- singletons into A, and that should do it... And then it was just a matter of making sure it worked. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Cartesian product non-associativity MoeBlee === Subject: Re: The meaning of set? > Hi All, The meaning of sets and set membership: Sets can be intuitively thought of as conceptual containers having > clear cut inclusion and exclusion rules. A container is what has the ability to contain some of what is other > than itself. The ability to contain is not determined by the existence > of contents within the container. The ability to contain is a de novo > property of the container itself. Intuitively speaking one can say that what makes a container able to > contain others is the concept of closure. For example this can be > shown well in geometric figures. > Geometric figures that can contain things within them should be closed > figures, open figures cannot contain things within them. This closure is what makes a geometric figure able to contain other > geometric figures inside it. However in the conceptual world, a concept is said to have closure if > it can hold a clear cut meaning, i.e. Dichotomous, in such a manner > that other concepts can be related to it in a clear cut manner. More rigorously speaking closure is having a clear cut inclusion > exclusion rules. Inclusion rule to a container is the requirement for being contained > within it. > Exclusion rule of a container is the requirement for being not > contained within it. Such containers if specified by their contents are called sets. Set membership is the inclusion rule to the set. A is a member of B mean that A fulfils the requirements of inclusion > into B. So a member of a set is what fulfils the requirements of inclusion into > that set. Simply speaking a member of a set is what fulfils its membership. > The Principle of conditional generality: For every set there is a set which specifically has it as its only > member, and every > member of a set is a set of some other sets, unless sets are defined in > such a manner as to make this rule logically contradictive. . > theory. 1) Extentiality and the empty set are very clearly derived from the > meaning of sets. > 2) Pairing: can be simply derived from the principle of conditional > generality as below. If x is a set and y is a set, then the container which specifically > contains them as its sole members is by definition a set since it is a > conceptual container having closure > As defined by a clear cut set membership/exclusion rule. 3) Union, separation, replacement, infinity, choice, power set. All can > simply be derived from the meaning of sets mentioned above as far as > they involve no logical contradiction. 4) The axiom of regularity is inherit in the meaning of sets as > conceptual containers of others. The set of all sets exist, but the set which contain it as a member > doesn't exist, nor does a proper superset of it exist, neither a power > set. Though axioms in ZFC are rigorously followed in a blind manner, yet > giving flexibility to them so that they are only applicable within in > the confines of logic, is a better and a more reasonable approach. Zuhair Abstract container is alright as the correct style of symbol for set. A cause to inclusion as the relation itself appears the difference of the set you use and common set theory. You is better than common theory except the fallacy of the red ball is always used to refute your theory of set. A set of All as abstracted red third order set is truely existenal as long as the relation of red exists. SO when the red ball container is made how does the inverted removal of balls occur? A rule change is required. So find the rule change to allow. I am out of time. it is a complex subject you have. Some people spend a lifetime of professional study and never understand the rather simple inverted ball. A red removal as the next, causes only red to be displayed out of the container. So true relative color has to be the cause of element in your theory. And this then begins the walk to the Aristotle translation book. Read it. And the set to be all as the abstract third order All element a. So I leave the comment with the abstract number dilemma displayed. All as abstracted set inclusive must have its number relative to the example set or the number can not relate all. And the Paradox of Zeno is this challenge. So it is a big thing to solve. === Subject: Re: The meaning of set? >> The good thing about ZFC is that we have an algorithm for generating >> all the sentences we accept as theorems. Really? What is that algorithm? Rupert didn't demand that the algorithm generate all and ONLY the sentences we accept as theorems. Lee Rudolph === Subject: Re: The meaning of set? > The good thing about ZFC is that we have an algorithm for generating > all the sentences we accept as theorems. >> Really? What is that algorithm? >Rupert didn't demand that the algorithm generate all and ONLY the > sentences we accept as theorems. Sure. But then an algorithm listing every sentence in the language of set theory does as well, and the relevance of the observation is even harder to see. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: The meaning of set? >Hi All, > The meaning of sets and set membership: > Sets can be intuitively thought of as conceptual containers having >clear cut inclusion and exclusion rules. > A container is what has the ability to contain some of what is other >than itself. The ability to contain is not determined by the existence >of contents within the container. The ability to contain is a de novo >property of the container itself. > Intuitively speaking one can say that what makes a container able to >contain others is the concept of closure. For example this can be >shown well in geometric figures. >Geometric figures that can contain things within them should be closed >figures, open figures cannot contain things within them. > This closure is what makes a geometric figure able to contain other >geometric figures inside it. > However in the conceptual world, a concept is said to have closure if >it can hold a clear cut meaning, i.e. Dichotomous, in such a manner >that other concepts can be related to it in a clear cut manner. > More rigorously speaking closure is having a clear cut inclusion >exclusion rules. > Inclusion rule to a container is the requirement for being contained >within it. >Exclusion rule of a container is the requirement for being not >contained within it. > Such containers if specified by their contents are called sets. > Set membership is the inclusion rule to the set. > A is a member of B mean that A fulfils the requirements of inclusion >into B. > So a member of a set is what fulfils the requirements of inclusion into >that set. > Simply speaking a member of a set is what fulfils its membership. > > The Principle of conditional generality: > For every set there is a set which specifically has it as its only >member, and every >member of a set is a set of some other sets, unless sets are defined in >such a manner as to make this rule logically contradictive. . > > theory. > 1) Extentiality and the empty set are very clearly derived from the >meaning of sets. >2) Pairing: can be simply derived from the principle of conditional >generality as below. > If x is a set and y is a set, then the container which specifically >contains them as its sole members is by definition a set since it is a >conceptual container having closure >As defined by a clear cut set membership/exclusion rule. > 3) Union, separation, replacement, infinity, choice, power set. All can >simply be derived from the meaning of sets mentioned above as far as >they involve no logical contradiction. > 4) The axiom of regularity is inherit in the meaning of sets as >conceptual containers of others. > The set of all sets exist, but the set which contain it as a member >doesn't exist, nor does a proper superset of it exist, neither a power >set. > Though axioms in ZFC are rigorously followed in a blind manner, yet >giving flexibility to them so that they are only applicable within in >the confines of logic, is a better and a more reasonable approach. > > Zuhair > Sorry, I'm not clear on what alternative foundation you're proposing. > Are you proposing a different axiomatic theory? What I am proposing is a different approach to sets and analysis based > on them. The ironic , congrete thinking like processing that is present in ZFC > and perhaps other set theories is yach, strangulating. I want a set theory that differentiates between what I call Necessary > characteristics for Identity and un-necessary characterisitcs. If Moe > Blee comes every day wearing a black trouseurs, it doesn't follow that > if one day he wears a white one then he is not Moe Blee. If water is evapourted, or freezed to make ice, it is still WATER. But that doesn't help any, because water is only water after freezing, because H2O has a well-defined triple point, which most liquids don't. Which is the same reason black trousers are isomorphic to white trousers, because dyes don't affect the geometric properties of trousers. But if I cleave one hydrogen itom from H20, then I will have )OH- and > that is not water. Unless Sets and set membership is clearly defined, no theory can > determine what characteristics are essential to identify x as a set. identity of a set, and these are the axiom of extentiality and the > axiom of regularity. the others are not essential characteristics of > sets. The other axioms are to be treated with Flexibility. >Zuhair === Subject: Re: The meaning of set? The good thing about ZFC is that we have an algorithm for generating > all the sentences we accept as theorems. Really? What is that algorithm? > Perhaps you interpreted me as saying there is an algorithm for testing whether a given sentence is a theorem. That's not true, of course. > -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: The meaning of set? The good thing about ZFC is that we have an algorithm for generating > all the sentences we accept as theorems. Really? What is that algorithm? > What I have in mind is an algorithm for listing every theorem of ZFC. Such an algorithm exists because this set is recursively enumerable. One possible algorithm would be one which searched through every possible string of symbols and determined whether or not it was a proof in ZFC, and if so, label the last line of the proof as a theorem. This algorithm would be non-terminating since it lists an infinite set of sentences, but every theorem would be verified as such within a finite amount of time. I'm not sure what your concern is. Perhaps you find my use of the phrase the sentences we accept as theorems to be imprecise. Or perhaps you don't agree with my calling this an algorithm since it is non-terminating. > -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: The meaning of set? > The good thing about ZFC is that we have an algorithm for generating > all the sentences we accept as theorems. >> Really? What is that algorithm? >What I have in mind is an algorithm for listing every theorem of ZFC. > Such an algorithm exists because this set is recursively enumerable. Sure. > I'm not sure what your concern is. Perhaps you find my use of the > phrase the sentences we accept as theorems to be imprecise. That was my concern. The recursive enumerability of the set of theorems of ZFC gives us no clue as to what algorithm it is, if any, that generates all the sentences we accept as theorems (in general, or of ZFC), unless we accept as theorems is intended to be read as for which formal derivations exist in ZFC. So I wonder what the relevance of your observation was, if it was just that the set of theorems of ZFC is recursively enumerable? Zuhair's different approach to sets and analysis based on them might well be pure waffle, as the case appears to be, but not because he doesn't present an algorithm for generating everything that he would accept as a theorem - whatever sense of accepting is meant here. We might just as well object that set theory is unacceptable because we can't given an explicit algorithm listing all the theorems acceptable on basis of the informal principles. What Zuhair needs to do is to present a coherent and interesting account of sets - he can worry about precise axiomatization, let alone formalization, later. I'll be holding my breath. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: The meaning of set? The good thing about ZFC is that we have an algorithm for generating > all the sentences we accept as theorems. >> Really? What is that algorithm? > What I have in mind is an algorithm for listing every theorem of ZFC. > Such an algorithm exists because this set is recursively enumerable. Sure. > I'm not sure what your concern is. Perhaps you find my use of the > phrase the sentences we accept as theorems to be imprecise. That was my concern. The recursive enumerability of the set of theorems > of ZFC gives us no clue as to what algorithm it is, if any, that > generates all the sentences we accept as theorems (in general, or of > ZFC), unless we accept as theorems is intended to be read as for > which formal derivations exist in ZFC. So I wonder what the relevance > of your observation was, if it was just that the set of theorems of ZFC > is recursively enumerable? > Perhaps recursive enumerability is not the main point. But I think it's important to have some precise characterization of what you accept as a sound argument and what you don't. Arguments in set theory are in principle formalizable, and then it is machine-checkable whether an argument is sound or not. So we have a precise characterization of what counts as correct reasoning and what doesn't. I was encouraging Zuhair to try to give a precise characterization of what he would be prepared to accept as a sound argument. > Zuhair's different approach to sets and analysis based on them might > well be pure waffle, as the case appears to be, but not because he > doesn't present an algorithm for generating everything that he would > accept as a theorem - whatever sense of accepting is meant here. We > might just as well object that set theory is unacceptable because we > can't given an explicit algorithm listing all the theorems acceptable on > basis of the informal principles. What Zuhair needs to do is to present > a coherent and interesting account of sets - he can worry about precise > axiomatization, let alone formalization, later. I'll be holding my breath. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: The meaning of set? > Perhaps recursive enumerability is not the main point. But I think it's > important to have some precise characterization of what you accept as a > sound argument and what you don't. Arguments in set theory are in > principle formalizable, and then it is machine-checkable whether an > argument is sound or not. So we have a precise characterization of what > counts as correct reasoning and what doesn't. We have a precise characterization of what counts as correct reasoning, though we have no precise characterization of what counts as an acceptable principle of set theory. In practice, the axioms of ZFC are sufficient, but an indefinite number of principles not provable in ZFC follow form the basic informal principles of set theory, although this observation isn't at all interesting in context of ordinary mathematics; also, in practice we almost never produce machine checkable proofs, only proofs we're sure could be formalized in ZFC in some idealized sense. > I was encouraging Zuhair > to try to give a precise characterization of what he would be prepared > to accept as a sound argument. That's certainly a good idea. Asking for an algorithm listing what he is prepared to accept as a sound argument doesn't strike me as a particularly useful way of doing that, however. Surely just lying down the basic principles and the basic ideas is sufficient. I haven't got the impression that he's trying to introduce a new logic, so evaluating his arguments - provided he's given a coherent explanation of his new conception of sets - is no different from evaluating arguments in mathematics in general. There's no particular need to go formal, unless we're actually interested in, say, proof theoretical properties of his own system. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Deriving a formula for the process queue size?? We have a process which processes the incoming messages and sends the responses out. This process has a queue associated with it. The handling is as follows: a) - Whenever the new message(Message 'A') arrives, if the process is free, Then The new message will be delivered to the process immediately -> Queue is empty if it is busy handling the already arrived message, then The new message will be stored into the queue. -> Queue starts getting filled - As soon as the process completes processing for the message that it has taken before(and becomes free now), the message from the queue will be taken out and processed by the process.. - Time taken for processing of each of these new incoming message 'A' is: t1 ms (I want to vary this and see the resulting queue size) b) - Apart from that, for each of the 80% of these incoming messages, there will be an additional message(message 'B') incoming to the queue.(Background: This is a message from the remote entity, for the response that was sent out from this process earlier. The remote entity takes 10 ms to send this message towards this process). - Time taken for processing of each of these additional message 'B' is: 3 ms c) - When the process is busy, both the normally incoming message and the additional incoming messages are stored in the queue. Then, they will be taken out and processed by the process in the order of their arrival. Now, how can we derive a formula for the queue size, for the given incoming message rate and at the given point in time? I tried to get, but I failed. Can anybody help? === Subject: Re: find intersection point of sphere and line > An ellipsoid is centered at the origin and its semi-axes are 3, 3, 5 in > the X, Y and Z directions respectively. A point light source with > intensity 200 is located at (6,5,0). > 1st) I have problem finding the point coordinate (x,y,z) that the light > will hit on the sphere. Hint:The bisector of focal rays at required point is normal to ellipse > section at z = 0. It passes through the given outside point. Why on xy plane? it is because of (6, 5, 0). === Subject: Generating Large Primes I've done a quick Google to look for a technique that I have in mind but I couldn't find anything (note: I did say quick). It seems to me that taking the product of the first n primes and adding 1 would be a good start. Certainly this number would not be divisible by any number up to and including the n-th prime. I don't know how to prove (or disprove?) that the resulting number cannot be divisible by any number greater than the n-th prime. Does anyone know if this particular technique has already been investigated? === Subject: Re: Generating Large Primes [Phil Milmoe] > I've done a quick Google to look for a technique that I have in mind > but I couldn't find anything (note: I did say quick). It seems to me > that taking the product of the first n primes The product of all primes less than or equal to prime p is called the primorial of p, and is usually denoted as p# For example, 2# = 2, 3# = 2*3 = 6, 5# = 2*3*5 = 30, and so on. Adding primorial to your search terms will get you better hits. > and adding 1 would be a good start. When p#+1 or p#-1 is prime, that's called a primorial prime. It's actually unusual for p# +/- 1 to be prime. Chris Caldwell has a nice summary here: http://primes.utm.edu/top20/page.php?id=5 > Certainly this number would not be divisible by any number up to > and including the n-th prime. Right. > I don't know how to prove (or disprove?) that the resulting number > cannot be divisible by any number greater than the n-th prime. 17 is the smallest prime p such that p#+1 and p#-1 are both composite. Above 17, that's the rule rather than the exception. > ... === Subject: Re: Generating Large Primes I found a way to generate quickly large primes: Donwload free, the program FACTOR.EXE of Shamus Software. http://primes.utm.edu/links/programs/large_arithmetic/ On the prompt write: factor -f xxxxxxxxxxxxxx*6 - 1 xxxxxxxxxxxx being a random integer with no more of 80 digits. (Do not to forget the -f ) Press ENTER . If it is not a prime , change -1 by +1 and check. Continue changing the sumand by 5,7,11,13,17,19,23,25,29,.....and checking. These numbers have the diferences : 2,4,2,4,2,4,2,4,2....... As the maximum difference between primes p(n+1) - p(n) if less than [log(p(n)]^2 the next prime is not far away. Ludovicus === Subject: Re: Generating Large Primes luiroto@yahoo ha escrito: > I found a way to generate quickly large primes: > Donwload free, the program FACTOR.EXE of Shamus Software. > http://primes.utm.edu/links/programs/large_arithmetic/ > On the prompt write: factor -f xxxxxxxxxxxxxx*6 - 1 > xxxxxxxxxxxx being a random integer with no more of 80 digits. > (Do not to forget the -f ) > Press ENTER . If it is not a prime , change -1 by +1 and check. > Continue changing the sumand by 5,7,11,13,17,19,23,25,29,.....and > checking. > These numbers have the diferences : 2,4,2,4,2,4,2,4,2....... > As the maximum difference between primes p(n+1) - p(n) if > less than [log(p(n)]^2 the next prime is not far away. > Ludovicus I forget to indicate that after seing the factors of the number, it is necessary to press F3 for retrieve the last number checked and then change the sumand. If the process of calculating the factors takes seconds, interrupt by presing CTRL BREAK , because the number is surely composite. === Subject: Re: Generating Large Primes > It seems to me that taking the product of the first n primes and adding 1 would be a good start. This doesn't work in practice as noted in the other posts. For techniques that do work, go here: === Subject: Re: Generating Large Primes > I've done a quick Google to look for a technique that I have in mind but I couldn't find anything (note: I did say quick). It seems to me that taking the product of the first n primes and adding 1 would be a good start. Certainly this number would not be divisible by any number up to and including the n-th prime. I don't know how to prove (or disprove?) that the resulting number cannot be divisible by any number greater than the n-th prime. Does anyone know if this particular technique has already been investigated? (I apologize if this is a repost - Google is being tempermental today) The following Derive-generated table gives the primality status of the first 100 numbers of the form 2*3*5 ... + 1: (The wierd formatting is an artifact of the Derive to clipboard to text transitions) ,, 1 prime ? ? ? ? 2 prime ? ? ? ? 3 prime ? ? ? ? 4 prime ? ? ? ? 5 prime ? ? ? ? 6 composite ? ? ? ? 7 composite ? ? ? ? 8 composite ? ? ? ? 9 composite ? ? ? ? 10 composite ? ? ? ? 11 prime ? ? ? ? 12 composite ? ? ? ? 13 composite ? ? ? ? 14 composite ? ? ? ? 15 composite ? ? ? ? 16 composite ? ? ? ? 17 composite ? ? ? ? 18 composite ? ? ? ? 19 composite ? ? ? ? 20 composite ? ? ? ? 21 composite ? ? ? ? 22 composite ? ? ? ? 23 composite ? ? ? ? 24 composite ? ? ? ? 25 composite ? ? ? ? 26 composite ? ? ? ? 27 composite ? ? ? ? 28 composite ? ? ? ? 29 composite ? ? ? ? 30 composite ? ? ? ? 31 composite ? ? ? ? 32 composite ? ? ? ? 33 composite ? ? ? ? 34 composite ? ? ? ? 35 composite ? ? ? ? 36 composite ? ? ? ? 37 composite ? ? ? ? 38 composite ? ? ? ? 39 composite ? ? ? ? 40 composite ? ? ? ? 41 composite ? ? ? ? 42 composite ? ? ? ? 43 composite ? ? ? ? 44 composite ? ? ? ? 45 composite ? ? ? ? 46 composite ? ? ? ? 47 composite ? ? ? ? 48 composite ? ? ? ? 49 composite ? ? ? ? 50 composite ? ? ? ? 51 composite ? ? ? ? 52 composite ? ? ? ? 53 composite ? ? ? ? 54 composite ? ? ? ? 55 composite ? ? ? ? 56 composite ? ? ? ? 57 composite ? ? ? ? 58 composite ? ? ? ? 59 composite ? ? ? ? 60 composite ? ? ? ? 61 composite ? ? ? ? 62 composite ? ? ? ? 63 composite ? ? ? ? 64 composite ? ? ? ? 65 composite ? ? ? ? 66 composite ? ? ? ? 67 composite ? ? ? ? 68 composite ? ? ? ? 69 composite ? ? ? ? 70 composite ? ? ? ? 71 composite ? ? ? ? 72 composite ? ? ? ? 73 composite ? ? ? ? 74 composite ? ? ? ? 75 prime ? ? ? ? 76 composite ? ? ? ? 77 composite ? ? ? ? 78 composite ? ? ? ? 79 composite ? ? ? ? 80 composite ? ? ? ? 81 composite ? ? ? ? 82 composite ? ? ? ? 83 composite ? ? ? ? 84 composite ? ? ? ? 85 composite ? ? ? ? 86 composite ? ? ? ? 87 composite ? ? ? ? 88 composite ? ? ? ? 89 composite ? ? ? ? 90 composite ? ? ? ? 91 composite ? ? ? ? 92 composite ? ? ? ? 93 composite ? ? ? ? 94 composite ? ? ? ? 95 composite ? ? ? ? 96 composite ? ? ? ? 97 composite ? ? ? ? 98 composite ? ? ? ? 99 composite ? ? ? ... 100 composite ? By the way, the 100th number is 4711930799906184953162487834760260422020574773 4096755201886348396164153358450342212052892567 0554468197243910409777715799180438028421831503 8719444943990492579030720635990538452312528339 864352999310398481791730017201031091 Derive balks upon being asked to factor it, so it probably doesn't have any small prime factors. -John Coleman === Subject: Re: Generating Large Primes > >> I've done a quick Google to look for a technique that I have in mind but > I couldn't find anything (note: I did say quick). It seems to me that > takin g the product of the first n primes and adding 1 would be a good > start. Certainly this number would not be divisible by any number up > to and including the n-th prime. I don't know how to prove (or > disprove?) that the resulting number cannot be divisible by any number > greater than the n-th prime. Does anyone know if this particular > technique has already been investigated? >(I apologize if this is a repost - Google is being tempermental today) >The following Derive-generated table gives the primality status of the > first 100 numbers of the form 2*3*5 ... + 1: >(The wierd formatting is an artifact of the Derive to clipboard to > text transitions) > Long table snipped >By the way, the 100th number is >4711930799906184953162487834760260422020574773 > 4096755201886348396164153358450342212052892567 > 0554468197243910409777715799180438028421831503 > 8719444943990492579030720635990538452312528339 > 864352999310398481791730017201031091 >Derive balks upon being asked to factor it, so it probably doesn't > have any small prime factors. >-John Coleman > Let X equal the product of the the first 100 primes plus one. X = 471193079990618495316248783476026042202057477340967552018863 4839616415335845034221205289256705544681972439104097777157991804 3802842183150387194449439904925790307206359905384523125283398643 52999310398481791730017201031091 X can be factored into a small prime, Y, and a 217 digit composite, Z. Y = 2879 Z = 163665536641409689238016249904837110872545146697105783959313 4713308932037459199104274153962037354873904980584959283486624454 4565071963581239039405849220189576348456533485718834013644806753 57068187008850917585973324429 === Subject: Re: Generating Large Primes > I've done a quick Google to look for a technique that I have in mind but > I couldn't find anything (note: I did say quick). It seems to me that > takin g the product of the first n primes and adding 1 would be a good > start. Certainly this number would not be divisible by any number up > to and including the n-th prime. I don't know how to prove (or > disprove?) that the resulting number cannot be divisible by any number > greater than the n-th prime. Does anyone know if this particular > technique has already been investigated? > (I apologize if this is a repost - Google is being tempermental today) > The following Derive-generated table gives the primality status of the > first 100 numbers of the form 2*3*5 ... + 1: > (The wierd formatting is an artifact of the Derive to clipboard to > text transitions) > Long table snipped > By the way, the 100th number is > 4711930799906184953162487834760260422020574773 > 4096755201886348396164153358450342212052892567 > 0554468197243910409777715799180438028421831503 > 8719444943990492579030720635990538452312528339 > 864352999310398481791730017201031091 > Derive balks upon being asked to factor it, so it probably doesn't > have any small prime factors. > -John Coleman >Let X equal the product of the the first 100 primes plus one. X = 471193079990618495316248783476026042202057477340967552018863 > 4839616415335845034221205289256705544681972439104097777157991804 > 3802842183150387194449439904925790307206359905384523125283398643 > 52999310398481791730017201031091 X can be factored into a small prime, Y, and a 217 digit > composite, Z. Y = 2879 Z = 163665536641409689238016249904837110872545146697105783959313 > 4713308932037459199104274153962037354873904980584959283486624454 > 4565071963581239039405849220189576348456533485718834013644806753 > 57068187008850917585973324429 Did you try factoring this composite directly? The FACTOR program from the MIRACL library often reports large composites that actually factor. === Subject: Re: Generating Large Primes > I've done a quick Google to look for a technique that I have in mind but > I couldn't find anything (note: I did say quick). It seems to me that > takin g the product of the first n primes and adding 1 would be a good > start. Certainly this number would not be divisible by any number up > to and including the n-th prime. I don't know how to prove (or > disprove?) that the resulting number cannot be divisible by any number > greater than the n-th prime. Does anyone know if this particular > technique has already been investigated? > (I apologize if this is a repost - Google is being tempermental today) > The following Derive-generated table gives the primality status of the > first 100 numbers of the form 2*3*5 ... + 1: > (The wierd formatting is an artifact of the Derive to clipboard to > text transitions) > Long table snipped > By the way, the 100th number is > 4711930799906184953162487834760260422020574773 > 4096755201886348396164153358450342212052892567 > 0554468197243910409777715799180438028421831503 > 8719444943990492579030720635990538452312528339 > 864352999310398481791730017201031091 > Derive balks upon being asked to factor it, so it probably doesn't > have any small prime factors. > -John Coleman >Let X equal the product of the the first 100 primes plus one. X = 471193079990618495316248783476026042202057477340967552018863 > 4839616415335845034221205289256705544681972439104097777157991804 > 3802842183150387194449439904925790307206359905384523125283398643 > 52999310398481791730017201031091 X can be factored into a small prime, Y, and a 217 digit > composite, Z. Y = 2879 Z = 163665536641409689238016249904837110872545146697105783959313 > 4713308932037459199104274153962037354873904980584959283486624454 > 4565071963581239039405849220189576348456533485718834013644806753 > 57068187008850917585973324429 Interesting. I just used Derive's built-in factors() function and killed the process after about a minute or so when I realized that (unless one were very lucky) there was little chance of finding a prime factorization for such a large number. Of course, as you point out, that state of affairs is consistent with the existence of small prime factors so I was a bit hasty when I said it probably doesn't have any small prime factors and should have hedged and just said probably doesn't factor into small primes. Derive is somewhat limited and doesn't have any built-in function which returns partial factorizations, although a crude one based on trial division would be easy enough to program. === Subject: Re: Generating Large Primes > I've done a quick Google to look for a technique that I have in mind but I couldn't find anything (note: I did say quick). It seems to me that taking the product of the first n primes and adding 1 would be a good start. Certainly this number would not be divisible by any number up to and including the n-th prime. I don't know how to prove (or disprove?) that the resulting number cannot be divisible by any number greater than the n-th prime. Does anyone know if this particular technique has already been investigated? 2*3*5*7*11*13+1=30031=59*509 2*3*5*7*11*13*17+1=510511=19*97*277 2*3*5*7*11*13*17*19+1=9699691=347*27953 2*3*5*7*11*13*17*19*23+1=223092871=317*703763 Jeroen === Subject: Re: Generating Large Primes <21944431.1161956639121.JavaMail.jakarta@nitrogen.mathforum.org>, Phil > I've done a quick Google to look for a technique that I have in mind but I > couldn't find anything (note: I did say quick). It seems to me that taking > the product of the first n primes and adding 1 would be a good start. > Certainly this number would not be divisible by any number up to and > including the n-th prime. I don't know how to prove (or disprove?) that the > resulting number cannot be divisible by any number greater than the n-th > prime. Does anyone know if this particular technique has already been > investigated? Yes, it has been investigated. 2*3*5*7*11*13+1 = 59*509 -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: algebra/geometry Sorry William but I cannot control the flow. Some unclear thoughts lead to others which might be clearer and so on. === Subject: Re: algebra/geometry days. My association with the Department is that of an alumnus. Cc: >Sorry William but I cannot control the flow. Some unclear thoughts lead to others which might be clearer and so on. Please: either learn to use Usenet properly, or stop using it. The two errors you keep committing: (1) QUOTE ENOUGH OF THE MESSAGE YOU ARE REPLYING TO IN ORDER TO PROVIDE CONTEXT. In MathForum, this is easily achieved by clicking on the Quote original button. And your second error. (2) FOLLOW UP TO THE MESSAGE YOU ARE REPLYING TO, not to any message you please. You have followed up on YOUR original post, instead of following up on the message you are replying to. Since, in addition, you fail to quote anything of the message you are replying to, it is impossible to figure out what the hell you are trying to say or to whom. So. Go to the post you want to follow up to, and click reply for ->THAT<- post, not for whichever one you please. And use the Quote original button. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: algebra/geometry Next time you might answer correctly Magidin. === Subject: Re: algebra/geometry days. My association with the Department is that of an alumnus. >Next time you might answer correctly Magidin. Well, you learned to follow up on the right message. But, apparently, you are still too dumb to click on the Quote original button. (Not to mention use commas to separate your clauses). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: algebra/geometry This question bothers for the mere fact that some teachers (those in my country) expect you to understand mathematics as if anything could be logically deducted from the connection of symbols. Some of these professors expect you to assume the veracity of mathematics by collecting end results from a perfectly well connected analysis. And by collecting bits of end results you can reach the conclusions. Conclusions which noone understands obviously. === Subject: Re: algebra/geometry === Subject: Re: algebra/geometry <7228059.1161938993311.JavaMail.jakarta@nitrogen.mathforum.org Following your comments it too much bother because you do not included context. Please do not do that. Reply only if adequate contexts is included _within_ the reply. http://oakroadsystems/genl/unice.htm#quote http://www.xs4all.nl/~hanb/documents/quotingguide.html Otherwise essential contexts is removed from view, the flow of thought disrupted and chaos reigns. === Subject: Re: Riemann geometry, chicken or the egg? > >> >> For example you might find a function that will produce the hex base >> digits of pi to infintity. You could calculate more hex digits of pi >> than are needed to describe anything in the physical universe. Is >> this example the same as developing a geometry that describes more >> htan 3 physical dimensions? Yes. No. I don't know. What do you mean? You can calculate pi to more digits than any physicist will ever need. >You can develop a geometry of four, five, or infinitely many >dimensions. Physicists use these all the time - Hilbert spaces are >(I'm told) hugely important in some parts of physics. Although I >suppose those extra dimensions aren't physical dimensions. What >do you mean by asking whether these two examples are the same? In any event, neither Riemann geometry, nor chickens & eggs, >presuppose more than three dimensions. > Clearly Reimann visualized more than three dimensions In a famous lecture he gave 10 June 1854, entitled On the Hypothesis That Lie at the Foundations of Geometry, Riemann emphasized that the truth about space is to be discovered not from perusal of the 2000-year-old books of Euclid but from physical experience. He pointed out that space could be highly irregular at very small distances and yet appear smooth at everyday distances. [Snip] === Subject: Re: Riemann geometry, chicken or the egg? >>I believe that doing geometry without an eye to physics goes back >to ancient Greece. Conic sections were studied pretty much for >their own sake. >>The motion of planets and asteroids were available to the greeks and >>it seems logical they might have wondered how you could describe an >>ellipse. Asteroids??? Asteroids and meteors streaking across the heavens were noted by the Greeks and Chinese . An observation such as this might beg the question why does this thing appear to be racing toward the sun? Of course it would take a second sighting of an asteroid or meteor streaking away from the sun before anyone could conceive of an elliptical orbit. The inspiration for experimenting with conical sections may have been something as simple as slicing a carrot who knows? >The Greeks knew seven planets (Sun, Moon, Mercury, Venus, Mars, >Jupiter, Saturn), but not asteroids. Some Greeks did have a >heliocentric system, but the idea that the orbits were ellipses >seems not to have occurred to anybody before Kepler. Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, Canada === Subject: Re: Riemann geometry, chicken or the egg? >Asteroids and meteors streaking across the heavens were noted by the > Greeks and Chinese . That meant they were in the atomosphere and burning up. So observing them (in that state) would not have yielded conic sections. Besides they were moving too fast for getting position fixes. Bob Kolker === Subject: Re: Riemann geometry, chicken or the egg? <8a20k25nr9ec167rgvgdr4mkrrsul6pe95@4ax> <7tj1k2h3iemc5m7iqnq8gn4lch1uft1gjk@4ax> ....................................... Some Greeks did have a > heliocentric system, but the idea that the orbits were ellipses > seems not to have occurred to anybody before Kepler. >Wiki -german about Alexander Thom: > Neben dem Ma¤system trug Thom auch zu einer tieferen Betrachtung der > Geometrie der megalithischen Steinkreise bei. Er klassifizierte sie in > sechs Typen: echte Kreise, Ellipsen, zwei Arten von eif.9armigen Kreisen > und zwei Arten von abgeflachten Kreisen. > The megalithic astronomers thought about orbits, of course there is no > hint in this to a heliocentric system. And more, an orbit with epicycles is not a circle. And there is one problem for astronomers and calendar-makers: count the days between of the four seasons between the equinoxes (day and night with equal length) and the winter and summer solstice, this does not divide the year into four equal parts. So, presumed sun is moving on a circular orbit around earth, it should have different speeds. So one will look for alternative orbits. Hero === Subject: Re: Character Tables for M24 and M12 > Making progress - >Could someone give me the character table for S4? > I understand the columns, and I am working to > understand the rows. An explanation of the 5 > different irreducible > representations in S4 would be nice >Well the obvious 4-dimensional representation 'theta' > (permutations > on the coordinates x1,x2,x3,x4) is easily seen to be > a sum of two > irreducible ones (compute the inner product of the > character with itself > - the same actually holds for any doubly transitive > group, but let's > not get into that yet). The vector x1+x2+x3+x4 is > stabilized by all of > S4, so the trivial character chi0 is a summand in > theta. The other > summand, say chi1, must then be an irreducible > 3-dimensional rep. Please show me chi1. Could you also show how theta (4 dim rep) is a sum of ch10 and chi1 >We also have the sign character. Let's call it chi2. > It is obviously > 1-dimensional. Theta((12))=2, as x3 and x4 are fixed > by (12), > so chi1((12))=2-chi0((12))=1. Therefore chi1*chi2 is > different from > chi1 (any transposition is an odd permutation). > Computing its > inner product with itself reveals that chi1*chi2 is > actually > irreducible, so we get another 3-dimensional > character chi3=chi1*chi2. Please show me chi3 >So far the squares of the dimensions add up to > 1+9+1+9=20, so we > have a 2-dimensional character missing. This is > gotten by using > a homomorphism f:S4->S3 and inflating the > 2-dimensional irreducible > character of S3. Or using another way, let theta' be > the character > of the conjugation action of S4 on the conjugacy > class of 3 > non-trivial elements of the elementary abelian > 2-group inside S4, namely > {y1=(12)(34),y2=(13)(24),y3=(14)(23)}. Again > (suitably interpreted) > y1+y2+y3 spans a trivial subrepresentation, and the > remaining > summand is the missing 2-dimensional character chi4. Is chi4 2-dimensional because it is based on double-permutations or because each perm is a two-swap (Sorry..) > NBTW chi4 vanishes on the odd permutations, so > tensoring this > representation with the sign representation doesn't > give you anything > new (you knew this beforehand, because we have > already accounted > for the 5 irreducible characters). >Hopefully this gets you started. I honestly think > that you should > be fully conversant with this type of exercises > before you try > your hand at M12 let alone M24. Glad you seem to have > realized this > yourself! I once did M12 to prove to myself that I > can do it. > Never tried M24. With suitable software at hand, it's > probably > a lot easier, but not nearly as much fun. >Enjoy! >Jyrki > === Subject: Re: Character Tables for M24 and M12 The vector x1+x2+x3+x4 is >> stabilized by all of >> S4, so the trivial character chi0 is a summand in >> theta. The other >> summand, say chi1, must then be an irreducible >> 3-dimensional rep. >Please show me chi1. Could you also show how theta (4 dim rep) is a sum of ch10 and chi1 theta(g) is simply the number of fixed points of the permutation g, so theta(1)=4, theta(12)=2, theta(123)=1, theta(1234)=theta((12)(34))=0 (the rest determined from these as I covered all the conjugacy classes). The sizes of the conjugacy classess (in the order above) are 1,6,8,6 and 3 respectively, so we can compute the inner product (theta,theta)=(1*4^2 + 6*2^2 + 8*1^2 + 9*0)/24 = 2. Therefore theta is a sum of two distinct irreducible characters. Obviously (theta,chi0)>0, so chi0 is one of them. The other is chi1 = theta - chi0, so (1) -> 3, (12) -> 1, (123) -> 0, (1234) -> -1, (12)(34) -> -1 character chi3=chi1*chi2. >Please show me chi3 Change the signs in the above as necessary. What's the problem? > Is chi4 2-dimensional because it is based on double-permutations or because each perm is a two-swap (Sorry..) No. The 3-dim rep with character theta' has chi0 as a component by the same argument as we used above with theta. So chi4 is 2-dimensional because 2=3-1. To get theta' do the following: Recall the notation y1=(12)(34), y2=(13)(24) and y3=(14)(23), and that the action of S4 on these is by conjugation - you do know that these 3 permutations form a conjugacy class? (1) centralizes all of y1, y2, y3, so theta'(1)=3, (12) centralizes y1, but swaps y2 and y3, so theta'(y1)=1, (123) permutes these in a 3-cycle, so theta'(123)=0, (1234) centralizes y2, but swaps y1 and y3, so theta'(1234)=1, (12)(34) centralizes all of y1, y2, y3, so theta'((12)(34))=3. Again (theta',theta')=(9+6+6+27)/24 = 2, so theta' is a sum of 2 distinct irreducible characters. Again (theta',chi0)>0, so chi0 is one of them. Subtract 1 from all these values to get chi4. There is a rather nice general character theory for the symmetric groups. I'm not saying that learning all of that is prerequisite to attacking Mathieu groups, but certainly the above computations shouldn't contain any mysteries. Jyrki === Subject: Re: Character Tables for M24 and M12 > Please show me chi1. Could you also show how theta > (4 dim rep) is a sum of ch10 and chi1 > theta(g) is simply the number of fixed points of the > permutation g, so > theta(1)=4, theta(12)=2, theta(123)=1, > theta(1234)=theta((12)(34))=0 > (the rest determined from these as I covered all the > conjugacy classes). > The sizes of the conjugacy classess (in the order > above) are 1,6,8,6 and > 3 respectively, so we can compute the inner product >(theta,theta)=(1*4^2 + 6*2^2 + 8*1^2 + 9*0)/24 = 2. >Therefore theta is a sum of two distinct irreducible > characters. > Obviously (theta,chi0)>0, so chi0 is one of them. The > other is > chi1 = theta - chi0, so (1) -> 3, (12) -> 1, (123) - 0, (1234) -> -1, > (12)(34) -> -1 >character chi3=chi1*chi2. >Please show me chi3 >Change the signs in the above as necessary. What's > the problem? * I'll look this up, is chi2 always the sign character? >Is chi4 2-dimensional because it is based on > double-permutations or because each perm is a > two-swap (Sorry..) * Sorry dumb question, now that I grasp the theory better > No. The 3-dim rep with character theta' has chi0 as a > component by the > same argument as we used above with theta. So chi4 is > 2-dimensional > because 2=3-1. To get theta' do the following: >Recall the notation y1=(12)(34), y2=(13)(24) and > y3=(14)(23), and > that the action of S4 on these is by conjugation - > you do know > that these 3 permutations form a conjugacy class? * Yes. How is it that this is chosen though (as theta')? > (1) centralizes all of y1, y2, y3, so theta'(1)=3, >(12) centralizes y1, but swaps y2 and y3, so > theta'(y1)=1, >(123) permutes these in a 3-cycle, so theta'(123)=0, >(1234) centralizes y2, but swaps y1 and y3, so > theta'(1234)=1, >(12)(34) centralizes all of y1, y2, y3, so > theta'((12)(34))=3. >Again >(theta',theta')=(9+6+6+27)/24 = 2, >so theta' is a sum of 2 distinct irreducible > characters. Again >(theta',chi0)>0, >so chi0 is one of them. >Subtract 1 from all these values to get chi4. >There is a rather nice general character theory for > the > symmetric groups. I'm not saying that learning all of > that > is prerequisite to attacking Mathieu groups, but > certainly > the above computations shouldn't contain any > mysteries. Jyrki > Paul Hjelmstad === Subject: derivative of an integral can anyone show me the steps for solving the following derivative of an integral: F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b find d F(x) / dx === Subject: Re: derivative of an integral > can anyone show me the steps for solving the following derivative of an > integral: F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b find d F(x) / dx As others have mentioned, your S_x_0 cos(xt)/t dt diverges. But if you consider instead G(x) = S_x_b cos(xt)/t dt, with 0 not in [x,b], then you should be able to show that G'(x) = (cos(b x) - 2 cos(x^2)) / x . David === Subject: Re: derivative of an integral >can anyone show me the steps for solving the following derivative of an >integral: F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b find d F(x) / dx > > First, an issue which consider important: One does not solve the derivative here. One solves an equation. Here we *determine* or *derive* a form of the derivative. As A.N. Niel pointed out, the integral does not converge. Let's consider instead F(x) = int(t=x..a, cos(x t) / t), where int denotes integral here, a is some positive number and x is positive as well. Note that F(x) = - int(t=a..x, cos(x t) / t). Letting g(u,x) = - int(t=a..u, cos(x t) / t), use the chain rule for functions of two variables to get F'(x) = D_1 g(u,x) du/dx + D_2 g(u,x) dx/dx where D_i is the ith partial derivative opertor and u = x. Then, by the Fundamental Theorem of Calculus and Leibniz's rule, F'(x) = - cos(x t) / t + int(t=a..x, sin(x t) / t), since u = x and du/dx = dx/dx =1. Interestingly, limit(F'(x), a ->0) does exist, which makes me think I might have made a mistake somewhere. Or maybe this is an example showing one cannot na.95vely switch limits and derivatives. If this is homework, cite sources in the submitted assignement. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: derivative of an integral > can anyone show me the steps for solving the following derivative of an >> integral: >> F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b >> find d F(x) / dx >> First, an issue which consider important: One does not solve the > derivative here. One solves an equation. Here we *determine* or > *derive* a form of the derivative. As A.N. Niel pointed out, the integral does not converge. Let's > consider instead > F(x) = int(t=x..a, cos(x t) / t), where int denotes integral > here, a is some positive number and x is positive as well. Note that > F(x) = - int(t=a..x, cos(x t) / t). Letting g(u,x) = - int(t=a..u, cos(x t) / t), use the chain rule for > functions of two variables to get F'(x) = D_1 g(u,x) du/dx + D_2 g(u,x) dx/dx where D_i is the ith partial derivative opertor and u = x. Then, by > the Fundamental Theorem of Calculus and Leibniz's rule, F'(x) = - cos(x t) / t + int(t=a..x, sin(x t) / t), D'oh! Now that I've read David Cantrell's response, boy do I see how I messed that up. It should be - cos(x^2) / x + int(t=a..x, sin(x t)), and you can take it from there. It is still the case that the limit exists as a approaches 0. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: derivative of an integral > can anyone show me the steps for solving the following derivative of an > integral: F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b find d F(x) / dx http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus === Subject: Re: derivative of an integral > can anyone show me the steps for solving the following derivative of an > integral: > F(x) = S_x_0 cos(xt)/t dt, where S_a_b is the integral from a to b > find d F(x) / dx >http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus > Notice the x inside the integral, as well as a limit of integration? You'll also need the chain rule for partial derivatives. And: the integral diverges at zero anyway... === Subject: result book hello, are there some results books such as the ones ramanujan used (i don't know the name of the ones he used, do you?) with just : definitions, lemmas, theorems, but no proofs... ? thankx ! a === Subject: Re: result book > hello, are there some results books such as the ones ramanujan used (i don't know the name > of the ones he used, do you?) with just : definitions, lemmas, theorems, but no proofs... ? Ramanujan used Formulas and theorems in pure mathematics, by George Shoobridge Carr. And dont' know if he used others besides this one. Here: http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html no other book of that kind is mentioned. Jose Carlos Santos === Subject: Grid Puzzle Given two parallel straight lines: q y - p x = 0 (1) q (y - b) - p (x - a) = 0 (2) Here (x,y) are real variables. But (p,q) and (a,b) are naturals. Question: for what values of a and b is the line (2) most close to the line (1), when given p and q ? Does there exist a closed formula? Or do we have to devise an algorithm? Attention shall be restricted to 0 < a < q and 0 < b < p . Example. Let 3 y - 2 x = 0 . Then the closest parallel lines to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , both resulting in a distance 1/sqrt(13) . Right? Hope I made no mistakes. It's Friday afternoon ... Han de Bruijn === Subject: Re: Grid Puzzle > Given two parallel straight lines: >q y - p x = 0 (1) >q (y - b) - p (x - a) = 0 (2) >Here (x,y) are real variables. > But (p,q) and (a,b) are naturals. >Question: for what values of a and b is the line (2) > most close to the line (1), when given p and q ? Does there > exist a closed formula? Or do we have to devise an algorithm? The distance between lines is proportional to abs(q b - p a ), which must be a non-negative integer. For a, b, p and q all positive integers, with variable integer values for a and b, the minimum of this will be the greatest common divisor of p and q, which can be found by the Euclidean algorithm, as can the values of a and b. See, for example, http://en.wikipedia.org/wiki/Euclidean_Algorithm === Subject: Re: Grid Puzzle Han de Bruijn skrev i melding > Given two parallel straight lines: q y - p x = 0 (1) q (y - b) - p (x - a) = 0 (2) Here (x,y) are real variables. > But (p,q) and (a,b) are naturals. Question: for what values of a and b is the line (2) > most close to the line (1), when given p and q ? Does there > exist a closed formula? Or do we have to devise an algorithm? Attention shall be restricted to 0 < a < q and 0 < b < p . Example. Let 3 y - 2 x = 0 . Then the closest parallel lines > to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , > both resulting in a distance 1/sqrt(13) . Right? Hope I made no mistakes. It's Friday afternoon ... Han de Bruijn Hi. A wild guess would be a = 1 and b = 1? Karl-Olav Nyberg === Subject: Re: Grid Puzzle Karl-Olav Nyberg skrev i melding Han de Bruijn skrev i melding >> Given two parallel straight lines: >> q y - p x = 0 (1) >> q (y - b) - p (x - a) = 0 (2) >> Here (x,y) are real variables. >> But (p,q) and (a,b) are naturals. >> Question: for what values of a and b is the line (2) >> most close to the line (1), when given p and q ? Does there >> exist a closed formula? Or do we have to devise an algorithm? >> Attention shall be restricted to 0 < a < q and 0 < b < p . >> Example. Let 3 y - 2 x = 0 . Then the closest parallel lines >> to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , >> both resulting in a distance 1/sqrt(13) . Right? >> Hope I made no mistakes. It's Friday afternoon ... >> Han de Bruijn Hi. A wild guess would be a = 1 and b = 1? Karl-Olav Nyberg No, no. Not so simple! Good question, I'll come back to it :) Karl-Olav Nyberg === Subject: Re: Grid Puzzle Karl-Olav Nyberg skrev i melding Karl-Olav Nyberg skrev i melding >> Han de Bruijn skrev i melding > Given two parallel straight lines: >> q y - p x = 0 (1) >> q (y - b) - p (x - a) = 0 (2) >> Here (x,y) are real variables. > But (p,q) and (a,b) are naturals. >> Question: for what values of a and b is the line (2) > most close to the line (1), when given p and q ? Does there > exist a closed formula? Or do we have to devise an algorithm? >> Attention shall be restricted to 0 < a < q and 0 < b < p . >> Example. Let 3 y - 2 x = 0 . Then the closest parallel lines > to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , > both resulting in a distance 1/sqrt(13) . Right? >> Hope I made no mistakes. It's Friday afternoon ... >> Han de Bruijn >> Hi. >> A wild guess would be a = 1 and b = 1? >> Karl-Olav Nyberg No, no. > Not so simple! Good question, I'll come back to it :) Karl-Olav Nyberg Hi again. As the variables involved are naturals, I stick to my first guess; a = 1 and b = 1, but there could be two lines. One upper and one lower. So my guess is abs(a) = 1 and abs(b) = 1. The distance between the two lines is given by: d = ((p / q) / sqrt(q^2 + p^2) * (pa-qb) Karl-Olav Nyberg === Subject: Re: Grid Puzzle Karl-Olav Nyberg skrev i melding Karl-Olav Nyberg skrev i melding >> Karl-Olav Nyberg skrev i melding >> Han de Bruijn skrev i melding >> Given two parallel straight lines: >> q y - p x = 0 (1) >> q (y - b) - p (x - a) = 0 (2) >> Here (x,y) are real variables. >> But (p,q) and (a,b) are naturals. >> Question: for what values of a and b is the line (2) >> most close to the line (1), when given p and q ? Does there >> exist a closed formula? Or do we have to devise an algorithm? >> Attention shall be restricted to 0 < a < q and 0 < b < p . >> Example. Let 3 y - 2 x = 0 . Then the closest parallel lines >> to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , >> both resulting in a distance 1/sqrt(13) . Right? >> Hope I made no mistakes. It's Friday afternoon ... >> Han de Bruijn >> Hi. >> A wild guess would be a = 1 and b = 1? >> Karl-Olav Nyberg >> No, no. >> Not so simple! >> Good question, I'll come back to it :) >> Karl-Olav Nyberg Hi again. As the variables involved are naturals, I stick to my first guess; a = 1 > and b = 1, but there could be two lines. One upper and one lower. So my > guess is abs(a) = 1 and abs(b) = 1. The distance between the two lines > is given by: d = ((p / q) / sqrt(q^2 + p^2) * (pa-qb) corr: d = ((p / q) / sqrt(q^2+p^2))*(pa-qb) Karl-Olav Nyberg === Subject: Re: Grid Puzzle > Karl-Olav Nyberg ... skrev ... > Karl-Olav Nyberg ... skrev ... >> Karl-Olav Nyberg ... skrev ... > Han de Bruijn ... skrev ... >> Given two parallel straight lines: >> q y - p x = 0 (1) >> q (y - b) - p (x - a) = 0 (2) >> Here (x,y) are real variables. >> But (p,q) and (a,b) are naturals. >> Question: for what values of a and b is the line (2) >> most close to the line (1), when given p and q ? Does there >> exist a closed formula? Or do we have to devise an algorithm? >> Attention shall be restricted to 0 < a < q and 0 < b < p . >> Example. Let 3 y - 2 x = 0 . Then the closest parallel lines >> to it are 3 (y - 1) = 2 (x - 1) and 3 (y - 1) = 2 (x - 2) , >> both resulting in a distance 1/sqrt(13) . Right? ... > As the variables involved are naturals, I stick to my first guess; a = 1 > and b = 1, but there could be two lines. One upper and one lower. So my > guess is abs(a) = 1 and abs(b) = 1. The distance between the two lines > is given by: ... > corr: d = ((p / q) / sqrt(q^2+p^2))*(pa-qb) To see that abs(a) = 1 and abs(b) = 1 is not a general solution, consider a simple variant of the earlier example, namely 6y - 4x = 0, ie, p=4, q=6. Take a=q/2=3, b=p/2=2; now 6(y-2) - 4(x-3) = 6y-4x. -jiw === Subject: expectation for number of throws of fair coin How one can compute expectation of number of throws for fair coin until you get certain pattern, say two heads in a row? I get some series as an solution which I can not sum up. But I know for example that the answer in last problem (for two heads) would 6. Could someone give me a hint how one solve such type of problems? === Subject: Re: expectation for number of throws of fair coin >How one can compute expectation of number of throws for fair coin until >you get certain pattern, say two heads in a row? >I get some series as an solution which I can not sum up. But I know for >example that the answer in last problem (for two heads) would 6. Could >someone give me a hint how one solve such type of problems? > > There are at least three methods, viz., Markov chains, renewal theory, and martingales. See Ross SM, Introduction to Probability Models, for example, for the first two approaches. Also see the thread, Monkey references to an earlier thread there. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: expectation for number of throws of fair coin <45427512.9090606@netscape.net> On Oct 27, 5:07 pm, Stephen J. Herschkorn How one can compute expectation of number of throws for fair coin until >you get certain pattern, say two heads in a row? >I get some series as an solution which I can not sum up. But I know for >example that the answer in last problem (for two heads) would 6. Could >someone give me a hint how one solve such type of problems?There are at least three methods, viz., Markov chains, renewal theory, > and martingales. See Ross SM, Introduction to Probability Models, for > example, for the first two approaches. Also see the thread, Monkey > references to an earlier thread there. -- > Stephen J. Herschkorn sjhersc...@netscape.net > Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: expectation for number of throws of fair coin > How one can compute expectation of number of throws for fair coin until > you get certain pattern, say two heads in a row? > I get some series as an solution which I can not sum up. But I know for > example that the answer in last problem (for two heads) would 6. Could > someone give me a hint how one solve such type of problems? Here is one possible method: Let E denote the expected number of tosses before you get two heads in a row. If your fist toss is a tails, then the expected number of total tosses before two heads is E + 1. If your first toss is a head and your second toss is a tail, then the expected number is E + 2. If you toss two heads, then the number of tosees is 2. These events occur with probability 1/2, 1/4, and 1/4, respectively, so E satisfies the equation E = 1 / 2 * (E + 1) + 1/ 4 * (E + 2) + 1 / 4 * (2), which yields E = 6. === Subject: M = ker(T) (+) im(T) Prove: If M is an R-Module and T a R-homomorphism from M to M, and if T^2 = T, then M = ker(T) (+) im(T), where (+) means direct sum. I can't seem to get started on this. I write out the definitions of what everything means but cannot make the next step. Any hint on how I should approach this? === Subject: Re: M = ker(T) (+) im(T) Originator: grubb@lola >Prove: If M is an R-Module and T a R-homomorphism from M to M, and if T^2 = T, then M = ker(T) (+) im(T), where (+) means direct sum. >I can't seem to get started on this. I write out the definitions of what everything means but cannot make the next step. Any hint on how I should approach this? If x is in M, what can you say about x-Tx? === Subject: Re: M = ker(T) (+) im(T) days. My association with the Department is that of an alumnus. >Prove: If M is an R-Module and T a R-homomorphism from M to M, and if T^2 = T, then M = ker(T) (+) im(T), where (+) means direct sum. I can't seem to get started on this. I write out the definitions of >what everything means but cannot make the next step. Any hint on how >I should approach this? First: can you show that ker(T) / im(T) = {0}? Let x in ker(T)/im(T). Then T(x)=0, and x = T(y) for some y in M. Now, since T^2 = T ... Then you want to prove that every element can be written as x = k+y, where k is in the kernel and y is in the image. Well, I know that T(T(z))=T(z) for every z. So... z - T(z) is in ker(T) for every z, yes? Does that suggest anything? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Percentage: $120 million is of $180 million? <4qcnedFmg68hU2@individual.net> <87wt6mg3ls.fsf@nonospaz.fatphil.org> But the newb did say paster, so who's confusing who? I don't know, maybe that's the Canadian spelling of the word, > like colour. No, it isn't. It is just incompetent misspelling, by somebody who spelling---just ask the OED. R.G. Vickson === Subject: Re: FLTMA: A little group theory <8764e84nmh.fsf@gmx.at> notes: (x,y,z) = 1 ==> Z/xZ* x Z/yZ* x Z/zZ* is cyclic. > No, that's too easy. ( x, y, z ) = 1 doesn't mean (phi(x), phi(y), > phi(z)) = 1. So it's not necesasrily cyclic, and we have only... For a direct product of groups to be cyclic, each factor must be cyclic, and we know that Z/xZ* is only cyclic in special cases. Note that Z/xZ x Z/yZ x Z/zZ is cyclic if and only if x,y,z are pairwise coprime, a stronger condition than (x,y,z) = 1. Proof: If Z/xZ x Z/yZ x Z/zZ is cyclic, then the projection map to say Z/xZ x Z/yZ must preserve a generator element, so Z/xZ x Z/yZ is also cyclic. But the order of an element (h,k) in Z/xZ x Z/yZ is the least common multiple of the order of h in Z/xZ and the order of k in Z/yZ, and must in particular be a divisor of LCM(x,y). Thus being cyclic requires LCM(x,y) = xy, which is equivalent to x and y coprime. Similarly for x and z, or for y and z. Conversely if x,y,z are pairwise coprime, then by the Chinese Remainder Thm. any system of equations: w = a mod x, w = b mod y, w = c mod z has a solution, i.e. w*(1,1,1) = (w,w,w) = (a,b,c) in Z/xZ x Z/yZ x Z/zZ so (1,1,1) is a generator for Z/xZ x Z/yZ x Z/zZ. QED === Subject: Re: FLTMA: A little group theory <8764e84nmh.fsf@gmx.at> We studied finitely generated abelian groups, direct products of > notes: > (x,y,z) = 1 ==> Z/xZ* x Z/yZ* x Z/zZ* is cyclic. > No, that's too easy. ( x, y, z ) = 1 doesn't mean (phi(x), phi(y), > phi(z)) = 1. So it's not necesasrily cyclic, and we have only... > For a direct product of groups to be cyclic, each factor must be > cyclic, and we know that Z/xZ* is only cyclic in special cases. Note that Z/xZ x Z/yZ x Z/zZ is cyclic if and only if x,y,z are > pairwise coprime, a stronger condition than (x,y,z) = 1. > (snip) Riiiiight. It is not common but not impossible for gcd(phi(x), phi(y), phi(z)) > 2, but it does happen. In no case, however, is this gcd = p. Hm. Phi(n) is even for all but the smallest n, and that was one thing that led me to this thread. We haven't even had Z/nZ* (factor groups?) yet, so I am writing ahead of myself. We have from a^n + b^n = c^n both x^p + y^p = z^p and x, y, z pairwise coprime, but Z/xZ x Z/yZ x Z/xZ is a cyclic, abelian, additive group, different from Z/xZ* x Z/yZ* x Z/zZ*, and this one is isomorphic to Z/(phi(x))Z x Z/(phi(y))Z x Z/(phi(z))Z, isn't it? This group may be isomorphic to a direct product of cyclic groups. I suppose we have those cases already somewhere, order = 2, 4, p^k, and 2p^k, I think it was, but then each factor must be cyclic. Groups of square free order are cyclic. Is there any way to get from x, y, z pairwise coprime and x^p + y^p = z^p to r, s, t, pairwise coprime and phi(r), phi(s), phi(t) square free, with r^p + s^p = t^p? I could do something with that, I think. I am fighting addiction to tobacco and near-addiction to caffeine now, and I'll tell you, it's one rainy day today. We are planning a vacation to get away from these habits. Does anyone here know a vacation spot (Saudi Arabia?) where tobacco and caffeine are not used or even available? Maybe I can move to Crystal City nearby, and work out an indoor lifestyle. You can't smoke indoors anywhere these days. :) It's clear that I am in some kind of cycle. A fairly stable one, with emphysema a likely result, but of little concern to the ADS providers, who are fighting fires daily. Doug === Subject: Re: FLTMA: A little group theory <8764e84nmh.fsf@gmx.at> We have gone from a^n + b^n = c^n to x^p + y^p = z^p with x,y,z coprime and a few other restrictions, then on to statements in modular arithmetic culminating in (z/y, z/x, x/y)^p = (1, 1, -1) in Z/xZ* x Z/yZ* x Z/zZ* and so the period of this group element is 2p, and < (z/y, z/x, x/y) > is a cyclic subgroup of the given group. p shares a factor of p with 2p and so | < (1,1,-1) > | = 2 again. If there is a prime q smaller than p it shares no factor with 2p and so generates the same cyclic subgroup. If there is no prime q smaller than p, and p = 2, and so by the lemma we haven't discussed much, but which seems clear to me, FLT follows. Now, can there be such a subgroup? I don't know yet. If (r,s,t) generates H, a subgroup of G, then (r,s,t)^(-1) generates the same subgroup. We can write ((y/z, x/z, y/x)^(-1))^p = (1,1,-1) by symmetry, I think, but can equal powers of presumably distinct (inverse) elements give the same value? I think not, unless every component value is self-inverse, so then z/y = y/z = zy = yz, etc., commuting. I am not sure of this last point, but we seem to be making some progress in exploring FLT with group theory and modular arithmetic. There is nothing here preventing a restart. That is, for any x, y, z, the direct product of their corresponding multiplicative groups exists, so that's an avenue for direct proof, rather than proof by contradiction, if I recall my proof styles correctly. Doug === Subject: To be happy in the temporal world and in the eternal abode :) hi everybody in this group i just want from everyone to visit these websites if he -she want to be happy in the temporal world and in the eternal abode :) discover Islam yourselves if u want there is hardly any place on earth today where Islam is totally unknown.More and more people have become curious enough to find out something about this much publicized religion ; more often than not, they have been pleasantly surprised. Islam is the religion and way of life of about one fifth of the world's population. Muslims are of diverse nationalities, cultures and races, but their religion teaches that all humanity is essentially equal. Islam is generally misunderstood and misrepresented in contemporary western societies; therefore, it is hoped that these websites will help shed light on Islam and dispel many of the prevailing misconceptions. http://www.islamtomorrow/default.asp http://www.missionislam/discover/why_they_became_muslim.htm http://mohammad.islamway/ http://www.harunyahya/index.php http://www.discover-islam.net/ http://www.islaam/ http://www.islam-qa/index.php?ln=eng http://www.55a.net/firas/english/index.php SONG http://saaid.net/flash/last-breath.htm May God guide us all to the truth. ^__^ === Subject: Convergent series If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) converges? === Subject: Re: Convergent series > If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) > converges? No. Consider the series 1/2^(1/3) - 1/2^(4/3) - 1/2^(4/3) + 1/3^(1/3) - 1/3^(4/3) - 1/3^(4/3) - 1/3^(4/3) ... + 1/n^(1/3) - 1/n^(4/3) - ... - 1/n^(4/3) ... Note that there are n subtractions in the nth row. This is a convergent series that sums to 0, but the sum of cubes diverges: In the nth row, after cubing, you get 1/n - 1/n^4 - ... - 1/n^4 = 1/n - 1/n^3, which is the nth term of a divergent series. === Subject: Re: Convergent series === Subject: Re: Convergent series === Subject: Re: Convergent series === Subject: Re: Convergent series I also tried to use Holder inequality, but I didn't found a proof (clearly, since the result is false!). This is not an homework, it was a curiosity of mine, but I think is a beautiful exercise in elementary series theory. === Subject: Re: Convergent series My first thought was that you could show the successive >> tails of the sequence of cubes approach zero as n --> oo >> by making use of Holder's inequality, maybe with p = 1/3 >> and q = 2/3, but I don't seem to be getting anywhere >> right now. [I did have an argument all typed up and >> ready to post. Then I thought about whether I should >> post it (since this is probably a homework problem), >> which got me to notice that my argument doesn't work.] > I also tried to use Holder inequality, but I didn't found > a proof (clearly, since the result is false!). This is > not an homework, it was a curiosity of mine, but I think > is a beautiful exercise in elementary series theory. Yes, I saw The World Wide Wade's post right after I sent mine in. What happened with my proof is that while I was deciding whether to post what I had written, or to change it so that I left the details out and just give the idea to try, I noticed that I had the inequality going the wrong way for what I wanted to conclude. Then, what was even worse, when I looked closer I realized that some of the computations weren't even correct. What I actually had wound up being an inequality of the form Z <= Z for a certain expression Z that involved the sum of the cubes of the terms in the tails (the square of this sum, I think). Dave L. Renfro === Subject: Re: Convergent series === Subject: Re: Convergent series > If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) > converges? > What were you able to do? How can you deduce convergence of one series from convergence of another? === Subject: Re: Convergent series <271020061317044392%anniel@nym.alias.net.invalid> A N Niel ha scritto: > If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) > converges? >What were you able to do? > How can you deduce convergence of one series from convergence of > another? I don't understand your answer. It is easy that if a_n >= 0 for every n than sum((a_n)^3) converges, and is also easy if sum(a_n) is a Leibniz series, but I don't know the general answer. I think it is yes, but I'm not sure. === Subject: Re: Convergent series > If a_n is such that sum(a_n) converges is it true > that sum((a_n)^3) converges? >> What were you able to do? >> How can you deduce convergence of one >> series from convergence of another? > I don't understand your answer. It is easy that if > a_n >= 0 for every n than sum((a_n)^3) converges, > and is also easy if sum(a_n) is a Leibniz series, > but I don't know the general answer. I think it is yes, > but I'm not sure. My first thought was that you could show the successive tails of the sequence of cubes approach zero as n --> oo by making use of Holder's inequality, maybe with p = 1/3 and q = 2/3, but I don't seem to be getting anywhere right now. [I did have an argument all typed up and ready to post. Then I thought about whether I should post it (since this is probably a homework problem), which got me to notice that my argument doesn't work.] Dave L. Renfro === Subject: Re: Convergent series > If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) > converges? > It would seem so, at least assuming all positive terms. The number of a_n greater than 1 must be finite if sum(a_n) converges, so the number of a_n^3 greater than 1 would be finite as well, and that portion converge. For all a_n less than or equal to 1, a_n^3 is less than or equal to a_n, and so that portion would converge as a series of lesser terms. I can see that with negative terms, sum(a_n^2) might not converge, but it seems like it would for a_n^3, for reasons like the above. I am sure someone has a more exact answer. Tony === Subject: Logrithmic spring.... sort of What I am trying to find/develop is a generalized formula with which I can specify the variables necessary to create a spiral on the surface of a cylinder whose RATE of revolutions-per-unit length changes. I tried to make that 'definition' as succinct as possible, so let me elaborate in hopes of clarifying what I'm after. If I were to take a cylinder of known length and diameter and wind a wire around it - like the rail on a spiral staircase - figuring out its length and its rise angle is fairly trivial. However, if I want that rise angle to decrease continuously, (twist rate increase), from the bottom of the staircase to the top, things get a bit more complicated... at least for me. There are other constraints: 1) At least for this first excercise, I want the curve, (staircase railing if you will), to make only one revolution regardless of the height of the cylinder, 2) Besides the diameter and height of the cylinder, the ultimate twist rate, (at the top of the stair), and the starting twist rate are the primary a priori input specifications. So, in specific example, let's say I start with a cylinder of: 1) 20 units length, and 2) 1/pi units in diameter, (circumference = 1), 3) a starting twist RATE of 1-revolution-per-100-units of height, 4) a final twist RATE of 1-revolution-per-10-units of height, and finally, 5) the spiral can make only one revolution from the base of the cylinder to the top. Ultimate questions: 1) If I 'unwrap' the surface of the cylinder so that it is a rectangle of unit width and 20 units length (height), what would the equation for the above-specified curve on the surface of the cylinder be? 2) I assume this curve is completely described by; cylinder height, cylinder diameter, starting AND ultimate 'twist rates', and the one-revolution-per-cylinder-length constraint. What I am interested in besides the basic equation, is whether or not the ratio of the starting and ending 'twist rates' can be specified instead of each actual value. In other words, can this equation be further generalized to four input parameters instead of five? Paul === Subject: Re: Logrithmic spring.... sort of > What I am trying to find/develop is a generalized formula with which I can specify the variables necessary to create a spiral on the surface of a cylinder whose RATE of revolutions-per-unit length changes. I tried to make that 'definition' as succinct as possible, so let me elaborate in hopes of clarifying what I'm after. If I were to take a cylinder of known length and diameter and wind a wire around it - like the rail on a spiral staircase - figuring out its length and its rise angle is fairly trivial. However, if I want that rise angle to decrease continuously, (twist rate increase), from the bottom of the staircase to the top, things get a bit more complicated... at least for me. There are other constraints: 1) At least for this first excercise, I want the curve, (staircase railing if you will), to make only one revolution regardless of the height of the cylinder, 2) Besides the diameter and height of the cylinder, the ultimate twist rate, (at the top of the stair), and the starting twist rate are the primary a priori input specifications. So, in specific example, let's say I start with a cylinder of: > 1) 20 units length, and > 2) 1/pi units in diameter, (circumference = 1), > 3) a starting twist RATE of 1-revolution-per-100-units of height, > 4) a final twist RATE of 1-revolution-per-10-units of height, and finally, > 5) the spiral can make only one revolution from the base of the cylinder to the top. Ultimate questions: 1) If I 'unwrap' the surface of the cylinder so that it is a rectangle of unit width and 20 units length (height), what would the equation for the above-specified curve on the surface of the cylinder be? 2) I assume this curve is completely described by; cylinder height, cylinder diameter, starting AND ultimate 'twist rates', and the one-revolution-per-cylinder-length constraint. What I am interested in besides the basic equation, is whether or not the ratio of the starting and ending 'twist rates' can be specified instead of each actual value. In other words, can this equation be further generalized to four input parameters instead of five? Paul Consider the development of a non-linear helix, wherein you have specified 4 boundary conditons. You can solve a fourth order differential equation as a boundary value problem with two conditions at start and and two at end.There is a very wide choice of differential equations. E.g., a simple one will be y'''' + y = 0 , y(0) = 0, y' (0) = 1/10, y(1) = 20, y'(1) = 1/100. Choose any suitable algebraic, trigonometric or logarithmic combination equation with 4 arbitrary constants.One way is to differentiate 4 times to form a DE and solve the boundary value problem. Another way is to solve it fully numerically. If you choose log curve like y = c Log( ax + b) with three constants DE will be of third order only and one condition will remain not satisfied. Narasimham === Subject: Functional system: f(x + f(x)) = f(x) + f(2x - f(x)) Is it possible to exist continuous R -> R function f such that for every x in R: f(x + f(x)) = f(x) + f(2x - f(x)) f(x) = x + f(-x) Theron === Subject: Re: Functional system: f(x + f(x)) = f(x) + f(2x - f(x)) >Is it possible to exist continuous R -> R function f such that for >every x in R: f(x + f(x)) = f(x) + f(2x - f(x)) >f(x) = x + f(-x) f(x) = max(x, 0) Mike Guy === Subject: abelian normalizer of a syl-p implies... Suppose that the normalizer N_G(P) of a p-Sylow subgroup P of the finite group G is abelian. Prove that G has a normal subgroup N such that G/N is isomorphic to P. Any hints are welcomed. By the way this is not HW. === Subject: Re: abelian normalizer of a syl-p implies... > Suppose that the normalizer N_G(P) of a p-Sylow subgroup P of the finite > group G is abelian. Prove that G has a normal subgroup N such that G/N is > isomorphic to P. Any hints are welcomed. By the way this is not HW. ************************************************************************ Hi: I don't know whether my last message passed on or not, so I'm resuming it again. Cehck the following: == Rotman's An intro. to the theory of groups, Transfer chapter. th. 7.50. == Robinson's A course in the theory of Groups, Transfer and etc., Theorem 10.18 (Burnside's), Chapter 10 == Rose's A course on Group Theory, Transfer and etc., 10.18... Tonio === Subject: Re: abelian normalizer of a syl-p implies... > Suppose that the normalizer N_G(P) of a p-Sylow subgroup P of the finite > group G is abelian. Prove that G has a normal subgroup N such that G/N is > isomorphic to P. Any hints are welcomed. By the way this is not HW. ****************************************************************** Hi: This seems to be a result by Burnside, since P is abelian (transfer homomorphism and following results: see Robinson's A course in the theory of groups, chapter 10. Check also 10.16 - 10.18 of Rose's A course on Group Theory. Also in Rotman's An intro. to the Theory of groups, lemma 7.44 , with theorem 7.50 being Burnside's Normal Complement Th.) Tonio === Subject: Re: abelian normalizer of a syl-p implies... > Suppose that the normalizer N_G(P) of a p-Sylow subgroup P of the finite > group G is abelian. Prove that G has a normal subgroup N such that G/N is > isomorphic to P. Any hints are welcomed. By the way this is not HW. ************************************************************************ Hi: Check the transfer theorems (Burnside) in the books on group theory by Rotman, Rose or Robinson. Tonio === Subject: Re: USA drives on right, Britian drives on left; Which is better, scientifically explored Because of the mechanics of bicycles and of precession, it is easier to >lean a well-aligned bicycle left than right. Stochastic variations in >balance will then cause a moving bicycle to tend left rather than >right. Okay, my current bicycle does tend to the left. But what breaks the symmetry? Torque-induced precession: http://hyperphysics.phy-astr.gsu.edu/hbase/bike.html If you take a wheel spinning on a horizontal axis forward in the sense that the front wheel of a bicycle spins, and apply a torque downwards on the left side of the axle (like leaning left) then the axis of rotation will precess to bring the left side of the axle back towards you, amplifying the tilt. Tom Davidson Richmond, VA === Subject: Quaternions (four-dimensional numbers) Regarding the multiplication of quaternions -- I originally learned it as i*k = j, k*i = -j; j*i = k, i*j = -k; and http://en.wikipedia.org/wiki/Quaternion has it described the other way around, with i*k equaling negative j and k*i equaling positive j. Do some people prefer it the first way? I'm pretty sure I once saw it explained in an algebra book that i*j, j*k, and k*i were the ones with the negative values. Can anyone explain what exactly the deal with this is? daniel mcgrath -- Daniel Gerard McGrath, a/k/a Govende: for e-mail replace invalid with com Developmentally disabled; has Autism (Pervasive Developmental Disorder), Obsessive-Compulsive Disorder, & periodic bouts of depression. [This signature is under construction.] === Subject: Re: Quaternions (four-dimensional numbers)- >Regarding the multiplication of quaternions -- I originally learned it as i*k = j, k*i = -j; j*i = k, i*j = -k; and >http://en.wikipedia.org/wiki/Quaternion has it described the other way >around, with i*k equaling negative j and k*i equaling positive j. Do >some people prefer it the first way? I'm pretty sure I once saw it >explained in an algebra book that i*j, j*k, and k*i were the ones with >the negative values. Can anyone explain what exactly the deal with >this is? daniel mcgrath > > The conventions i*j = k etc and j*i = k etc are transformed into each other by quaternion conjugation. The issue reminds me of a joke my teacher of Latin and Greek told us in the classroom: The Iliad and Odyssey were not written by Homer, but by another poet also named Homer.. Johan E. Mebius === Subject: Re: Quaternions (four-dimensional numbers) > Regarding the multiplication of quaternions -- >I originally learned it as i*k = j, k*i = -j; j*i = k, i*j = -k; and > http://en.wikipedia.org/wiki/Quaternion has it described the other way > around, with i*k equaling negative j and k*i equaling positive j. Do > some people prefer it the first way? I'm pretty sure I once saw it > explained in an algebra book that i*j, j*k, and k*i were the ones with > the negative values. Can anyone explain what exactly the deal with > this is? Consider that the basis vectors 1, i, j, k have in themselves no meaning apart from the quaternion division algebra. The form I learned [Herstein, _Topics in Algebra_, 1964] with the multiplication table x 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1 can be written with J=k and K=j as x 1 i J K 1 1 i J K i i -1 -K J J J K -1 -i K K -J i -1 which corresponds to your form. In either form, 1) ii = jj = kk = ijk = -1 2) ij = -ji 3) jk = -kj 4) ki = -ik The more common form has the advantage that (2-4) are all positive, and the cycle ij (=k), jk (=i), ki (-j) seems intuitive. In your form, the positive products are ik (=j), kj (=i), and ji (=k), an order which doesn't have alphabetization to speak for it. In any case, there is little reason other than convention to lead one to prefer one of these equivalent forms over the other. === Subject: Re: Quaternions (four-dimensional numbers) > Regarding the multiplication of quaternions -- >I originally learned it as i*k = j, k*i = -j; j*i = > k, i*j = -k; and > http://en.wikipedia.org/wiki/Quaternion has it > described the other way > around, with i*k equaling negative j and k*i > equaling positive j. Do > some people prefer it the first way? I'm pretty > sure I once saw it > explained in an algebra book that i*j, j*k, and k*i > were the ones with > the negative values. Can anyone explain what > exactly the deal with > this is? >Consider that the basis vectors 1, i, j, k have in > themselves no meaning > apart from the quaternion division algebra. The form > I learned > [Herstein, _Topics in Algebra_, 1964] with the > multiplication table > x 1 i j k > 1 1 i j k > i i -1 k -j > j j -k -1 i > k k j -i -1 >can be written with J=k and K=j as > x 1 i J K > 1 1 i J K > i i -1 -K J > J J K -1 -i > K K -J i -1 >which corresponds to your form. >In either form, > 1) ii = jj = kk = ijk = -1 > 2) ij = -ji > 3) jk = -kj > 4) ki = -ik >The more common form has the advantage that (2-4) are > all positive, and > the cycle ij (=k), jk (=i), ki (-j) seems intuitive. > In your form, the positive products are ik (=j), kj > (=i), and ji (=k), > an order which doesn't have alphabetization to speak > for it. In any > case, there is little reason other than convention to > lead one to prefer > one of these equivalent forms over the other. > It is interesting to speculate that Hamilton recognized that k*j*i = -1 would work just as well, and therefore he carved i*j*k = -1 in stone so that there was a clear president. - MO === Subject: Re: Quaternions (four-dimensional numbers) >there was a clear president. We could sure use a clear president now, whether he uses Bart -- The man without a .sig === Subject: Re: Quaternions (four-dimensional numbers) > Regarding the multiplication of quaternions -- >I originally learned it as i*k = j, k*i = -j; j*i = > k, i*j = -k; and > http://en.wikipedia.org/wiki/Quaternion has it > described the other way > around, with i*k equaling negative j and k*i equaling > positive j. Do > some people prefer it the first way? I'm pretty sure > I once saw it > explained in an algebra book that i*j, j*k, and k*i > were the ones with > the negative values. Can anyone explain what exactly > the deal with > this is? >daniel mcgrath > -- > Daniel Gerard McGrath, a/k/a Govende: > for e-mail replace invalid with com >Developmentally disabled; > has Autism (Pervasive Developmental Disorder), > Obsessive-Compulsive Disorder, > & periodic bouts of depression. > [This signature is under construction.] > Hamilton invented quaternions. He carved the following equation in stone: i*i = j*j = k*k = i*j*k = -1 That is the standard and only definition that I have seen in dozens of math books. See http://mathworld.wolfram/Quaternion.html - MO === Subject: Re: Quaternions (four-dimensional numbers) <27579597.1161970795190.JavaMail.jakarta@nitrogen.mathforum.org Hamilton invented quaternions. He carved the following equation in stone: i*i = j*j = k*k = i*j*k = -1 That is the standard and only definition that I have seen in dozens of math books. You need to add that it is a real associative algebra with basis {1,i,j,k}. Associativity is perhaps deducible from i*j*k = -1, since then you have an interpretation of i*j*k. Given associativity, products of i, j and k have a unique meaning, and we can multiply i*j*k on the left and the right and discover finally that quaterions are a skew field. === Subject: Re: Quaternions (four-dimensional numbers) days. My association with the Department is that of an alumnus. >Regarding the multiplication of quaternions -- I originally learned it as i*k = j, k*i = -j; j*i = k, i*j = -k; and >http://en.wikipedia.org/wiki/Quaternion has it described the other way >around, with i*k equaling negative j and k*i equaling positive j. They end up being isomorphic, of course. > Do >some people prefer it the first way? Most people use it the other way: i*j = k, j*k = i, k*i=j. That way, the first equation is in alphabetical order. > I'm pretty sure I once saw it >explained in an algebra book that i*j, j*k, and k*i were the ones with >the negative values. Can anyone explain what exactly the deal with >this is? The deal is that it doesn't matter, so long as you are consistent. Let us assume you want to define quaternions your way, with i*k = j, j*i=k, and k*j=i. Every quaternion can be written uniquely as a + b*i + c*j + d*k. Okay: I'm going to define J to be shorthand for -j. So 3 + 4*i -5*J + k means 3 + 4*i -5(-j) + k = 3 + 4*i + 5j + k. Now, clearly, every quaternion can be written uniquely in terms of i, J, and k, just as it can in terms of i, j, and k. And multiplication will now obey the following rules: i*J = i*(-j) = -(i*j) = -(-k) = k. J*i = (-j)*i = -(j*i) = -(k) = -k. J*k = (-j)*k = -(j*k) = -(-i) = i. k*J = k*(-j) = -(k*j) = -(i) = -i. i*k = j = -J. k*i = -j = J. so the multiplication table for i, J, k is just the other possible definition. It is also pretty easy to see that if you let x be any quaternion written in terms of i, j, and k; and you let f(x) be any quaternion written in terms of i, J, and k, then f(x+y) = f(x) + f(y) f(xy) = f(x)f(y) so that anything you do using i, j, and k, you get the same thing (with a flipped sign) if you use i, J, and k instead. So this means that you can define the multiplication either way, and everything will be essentially the same. It is just a matter of convenience. And, as I mentioned, as far as I know most people find the definition i*j = k more convenient. (You could do the same thing replacing i with I=-i, or k with K=-k, instead of replacing j). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Interesting problem (limits theory) Hello! I was studing a quite interesting problem for a few days and I have some possible solutions but nothing I'm really certain of. I was wondering if any of you have some certain (it means - mathematicaly correct) ideas how to solve the following puzzle: PROBLEM: Let f be a continuous real-valued function of a rational variable. Prove that there always exists an irrational number t in (RQ), such that there is a real number (it means - finite) limit of f(x) as x tends to t. We can also write it shortly as: Let f: Q->R, continuous. Show that there exists t in (RQ), that: lim f(x) is real as (x -> t). I would be very glad for your help, guys. And sorry for my poor English, but I'm not a native speaker. Chris precarion@yahoo.ca === Subject: Re: Interesting problem (limits theory) What topology do you use for Q? Without that piece of information, the problem is not well defined. > Hello! I was studing a quite interesting problem for a few days and I have some possible solutions but nothing I'm really certain of. I was wondering if any of you have some certain (it means - mathematicaly correct) ideas how to solve the following puzzle: PROBLEM: Let f be a continuous real-valued function of a rational variable. Prove that there always exists an irrational number t in (RQ), such that there is a real number (it means - finite) limit of f(x) as x tends to t. We can also write it shortly as: Let f: Q->R, continuous. Show that there exists t in (RQ), that: lim f(x) is real as (x -> t). I would be very glad for your help, guys. And sorry for my poor English, but I'm not a native speaker. >Chris >precar...@yahoo.ca === Subject: Re: Interesting problem (limits theory) > What topology do you use for Q? Without that piece of information, the > problem is not well defined. > I am assuming the topology defined on Q is the induced topology, i.e. the open set of Q is the intersection of Q and the open set of R . Hint: use Baire category theorem. Let B_(N,n) be a set of this kind (N, n are positive integers) : {t is in R: there exists t_1, t_2 in Q, such that |t_1 - t| < 1/N, |t_2 - t| < 1/N, and |f(t_1)- f(t_2)| >= 1/n}, let A_n be the intersection of all B_(N,n) , N runs from 1 to infinity. Then show that the closure of A_n is nowhere dense in R, thus RQ can not be the union of all A_n's, since the set of irrational numbers is of second category. The union of all A_n is a proper subset of the set of irrational numbers. Let C be the set: {r is in R: lim f(x) doesn't exist as x in Q goes to r }, then C is a subset of the union of all A_n's. === Subject: Re: Interesting problem (limits theory) What topology do you use for Q? Without that piece of information, the problem is not well defined. > Hello! I was studing a quite interesting problem for a few days and I have some possible solutions but nothing I'm really certain of. I was wondering if any of you have some certain (it means - mathematicaly correct) ideas how to solve the following puzzle: PROBLEM: Let f be a continuous real-valued function of a rational variable. Prove that there always exists an irrational number t in (RQ), such that there is a real number (it means - finite) limit of f(x) as x tends to t. We can also write it shortly as: Let f: Q->R, continuous. Show that there exists t in (RQ), that: lim f(x) is real as (x -> t). I would be very glad for your help, guys. And sorry for my poor English, but I'm not a native speaker. >Chris >precar...@yahoo.ca === Subject: Tough trig problem Does anyone see an easy solution to the following trig problem? If on a triangle ABC the equality B=2*A holds, then b^2-a^2=a*c, where a,b,c are the triangle sides. The student is supposed to use the sine and cosine laws, but I don't see it after 1 hour of looking. Part i) of the problem requires under the same assumption to prove that cos(A)=b/(2*a). I was able to get this from: a/sin(A)=b/sin(B) => a/sin(A)=b/(2*sin(A)*cos(A)) => a=b/(2*cos(A)) But part ii) has me stumped. -- ------- BACHELOR: A man who never makes the same mistake once. === Subject: Re: Tough trig problem escribi.97: > Does anyone see an easy solution to the following trig problem? If on a triangle ABC the equality B=2*A holds, then b^2-a^2=a*c, > where a,b,c are the triangle sides. The student is supposed to use the sine and cosine laws, but I don't > see it after 1 hour of looking. Part i) of the problem requires under the same assumption to prove > that cos(A)=b/(2*a). I was able to get this from: a/sin(A)=b/sin(B) = a/sin(A)=b/(2*sin(A)*cos(A)) = a=b/(2*cos(A)) But part ii) has me stumped. > b^2 = a^2 + c^2 - 2a*c*cos(B) = a^2 + c^2 - 2a*c*(2cos^2(A) - 1) = a^2 + c^2 - 2a*c*(2(b/(2a))^2 - 1) = a^2 + c^2 - b^2*c/a + 2a*c b^2(1 + c/a) = (a + c)^2 b^2(a + c)/a = (a + c)^2 b^2 = a(a + c) = a^2 + a* b^2 - a^2 = a*c -- Saludos, Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r === Subject: Re: Tough trig problem > b^2 = a^2 + c^2 - 2a*c*cos(B) = a^2 + c^2 - 2a*c*(2cos^2(A) - 1) = a^2 + c^2 - 2a*c*(2(b/(2a))^2 - 1) = a^2 + c^2 - b^2*c/a + 2a*c > b^2(1 + c/a) = (a + c)^2 b^2(a + c)/a = (a + c)^2 b^2 = a(a + c) = a^2 + a* b^2 - a^2 = a*c > -- > Saludos, Ignacio Larrosa Ca.98estro -- ------- Beam me up Scotty. This isn't the men's room. === Subject: Re: Tough trig problem Ignacio Larrosa Ca.96estro escribi.97: >> Does anyone see an easy solution to the following trig problem? >> If on a triangle ABC the equality B=2*A holds, then b^2-a^2=a*c, >> where a,b,c are the triangle sides. >> The student is supposed to use the sine and cosine laws, but I >> don't >> see it after 1 hour of looking. >> Part i) of the problem requires under the same assumption to prove >> that cos(A)=b/(2*a). >> I was able to get this from: >> a/sin(A)=b/sin(B) => a/sin(A)=b/(2*sin(A)*cos(A)) => a=b/(2*cos(A)) >> But part ii) has me stumped. b^2 = a^2 + c^2 - 2a*c*cos(B) = a^2 + c^2 - 2a*c*(2cos^2(A) - 1) = a^2 + c^2 - 2a*c*(2(b/(2a))^2 - 1) = a^2 + c^2 - b^2*c/a + 2a*c > b^2(1 + c/a) = (a + c)^2 b^2(a + c)/a = (a + c)^2 b^2 = a(a + c) = a^2 + a* b^2 - a^2 = a*c Yes. Or equivalently: In *any* triangle c = a*cos(B) + b*cos(A) Putting cos(B) = 2*cos^2(A)-1, and then cos(A) = b/(2*a) gives the required result. -- ?ref=61771 === Subject: Re: To simplify a product of 16 sines in 5 variables Hi ! Let SQRs.i denote sin(a.i)^2, let m[n1,n2,...] denote sum of all monomials in SQRs1,SQRs2,SQRs3,SQRs4, (not SQRs5) with degree n1,n2,... then the product can be expressed as ANS. ANS := (-2016*m[3,3,2]-256*m[6,2,1]-128*m[4,3,2]-40*m[6,1]-344*m[4,2,1]+176*m[ 5,1,1]-928*m[3,2,1,1]+752*m[4,1,1,1]-1984*m[5,3,1,1]+18176*m[4,2,2,2]-240*m[ 5, 3]-16*m[7,1]+96*m[6,2]+144*m[6,1,1]+1248*m[4,2,2]+320*m[4,4]-40*m[4,3]-160*m [3 ,2,2,1]-3232*m[4,3,1,1]+2976*m[5,2,1,1]-64*m[7,1,1,1]-12288*m[3,3,3,3]+1536* m[ 5,3,2,1]-4096*m[4,4,2,1]+2560*m[4,3,3,1]-1024*m[5,3,3,1]+5120*m[3,2,2,2]-883 2* m[2,2,2,2]-8192*m[4,3,2,2]-11264*m[3,3,2,2]+18432*m[3,3,3,2]+2544*m[4,2,1,1] + 384*m[3,3,1,1]+4096*m[4,4,1,1]+4352*m[4,3,2,1]-2880*m[5,1,1,1]-2304*m[5,2,2, 1] +224*m[6,1,1,1]+5056*m[3,3,2,1]-8320*m[4,2,2,1]+4096*m[4,4,2,2]-16000*m[3,3, 3, 1]+736*m[5,3,1]-768*m[5,3,2]+512*m[5,3,3]+256*m[6,2,2]-336*m[5,2,1]+4928*m[3 ,3 ,3]-1024*m[4,4,1]+272*m[3,2,2]+1024*m[4,4,2]-1280*m[4,3,3]+32*m[7,1,1]+208*m [4 ,3,1]+384*m[5,2,2]+72*m[5,2]+1520*m[2,2,2,1]+8*m[7]+416*m[3,3,1])*SQRs5-16*m [3 ,3,2]+96*m[6,2,1]+192*m[4,3,2]+(-3072*m[3,3,2]-128*m[4,2,1]+736*m[5,1,1]+384 *m [3,1,1,1]+5056*m[3,2,1,1]-3232*m[4,1,1,1]-1024*m[5,3,1,1]+8192*m[4,2,2,2]-25 6* m[5,3]-160*m[2,2,1,1]+2048*m[4,2,2]+512*m[4,4]-384*m[4,3]+416*m[3,1,1]-40*m[ 4, 1]-11264*m[3,2,2,1]+2560*m[4,3,1,1]+1536*m[5,2,1,1]-2016*m[3,2,1]-16384*m[3, 3, 3,3]-240*m[5,1]+38912*m[3,2,2,2]-38400*m[2,2,2,2]-928*m[2,1,1,1]-36864*m[3,3 ,2 ,2]+24576*m[3,3,3,2]+4352*m[4,2,1,1]-16000*m[3,3,1,1]+2112*m[2,2,2]-1984*m[5 ,1 ,1,1]+18432*m[3,3,2,1]-8192*m[4,2,2,1]-12288*m[3,3,3,1]+512*m[5,3,1]-768*m[5 ,2 ,1]+2048*m[3,3,3]+512*m[3,2,2]+272*m[2,2,1]-1280*m[4,3,1]+96*m[3,3]+208*m[4, 1, 1]+384*m[5,2]-16*m[3,2]+192*m[4,2]+5120*m[2,2,2,1]+56*m[5]+4928*m[3,3,1])* SQRs5^3+(1536*m[3,2,1,1]-2304*m[2,2,1,1]+736*m[3,1,1]-768*m[3,2,1]-336*m[2,1 ,1 ]+176*m[1,1,1]+2976*m[2,1,1,1]+72*m[2,1]+768*m[2,2,2]-1984*m[3,1,1,1]+384*m[ 2, 2,1]-240*m[3,1]-288*m[2,2]-256*m[3,3]+384*m[3,2]+512*m[3,3,1]-2880*m[1,1,1,1 ]+ 56*m[3]-1024*m[3,3,1,1])*SQRs5^5+736*m[5,3,1,1]-5152*m[4,2,2,2]+56*m[5,3]+8* m[ 7,1]-28*m[6,2]+(1024*m[4,2,1]-3232*m[3,1,1,1]+4352*m[3,2,1,1]+4096*m[4,1,1,1 ]- 4096*m[4,2,2,2]-8320*m[2,2,1,1]-1024*m[4,2,2]-256*m[4,4]+512*m[4,3]+208*m[3, 1, 1]+320*m[4,1]-70*m[4]-8192*m[3,2,2,1]+752*m[1,1,1,1]-128*m[3,2,1]-344*m[2,1, 1] +8192*m[3,2,2,2]-22272*m[2,2,2,2]+2544*m[2,1,1,1]-4096*m[4,2,1,1]+2560*m[3,3 ,1 ,1]-5152*m[2,2,2]+4096*m[4,2,2,1]+2048*m[3,2,2]+1248*m[2,2,1]-40*m[3,1]-36*m [2 ,2]-384*m[3,3]-1024*m[4,1,1]+192*m[3,2]-576*m[4,2]+18176*m[2,2,2,1]-1280*m[3 ,3 ,1])*SQRs5^4-40*m[6,1,1]-36*m[4,2,2]+(8*m[1]-16*m[1,1]+32*m[1,1,1]-64*m[1,1, 1, 1])*SQRs5^7+512*m[4,4,3]-70*m[4,4]+272*m[3,2,2,1]+208*m[4,3,1,1]-336*m[5,2,1 ,1 ]+32*m[7,1,1,1]-256*m[6,2,1,1]+768*m[5,2,2,2]+2048*m[3,3,3,3]+256*m[6,2,2,1] - 768*m[5,3,2,1]+1024*m[4,4,2,1]-1280*m[4,3,3,1]+512*m[5,3,3,1]+2112*m[3,2,2,2 ]- 2008*m[2,2,2,2]+2048*m[4,3,2,2]+512*m[3,3,2,2]-3072*m[3,3,3,2]-344*m[4,2,1,1 ]+ 416*m[3,3,1,1]-1024*m[4,4,1,1]-m[8]-128*m[4,3,2,1]+176*m[5,1,1,1]+384*m[5,2, 2, 1]+144*m[6,1,1,1]-2016*m[3,3,2,1]+1248*m[4,2,2,1]-1024*m[4,4,2,2]-256*m[6,2, 2, 2]+4928*m[3,3,3,1]+(512*m[3,3,2]+256*m[6,2,1]+2048*m[4,3,2]+96*m[6,1]+1248*m [4 ,2,1]-336*m[5,1,1]-928*m[3,1,1,1]-160*m[3,2,1,1]+2544*m[4,1,1,1]+1536*m[5,3, 1, 1]-22272*m[4,2,2,2]+384*m[5,3]-96*m[6,2]+1520*m[2,2,1,1]-256*m[6,1,1]-5152*m [4 ,2,2]-576*m[4,4]+192*m[4,3]+5120*m[3,2,2,1]+4352*m[4,3,1,1]-2304*m[5,2,1,1]+ 272*m[3,2,1]+24576*m[3,3,3,3]+4096*m[4,4,2,1]+72*m[5,1]-28*m[6]-38400*m[3,2, 2, 2]+48768*m[2,2,2,2]+8192*m[4,3,2,2]+38912*m[3,3,2,2]-36864*m[3,3,3,2]-8320*m [4 ,2,1,1]+5056*m[3,3,1,1]-4096*m[4,4,1,1]-2008*m[2,2,2]-8192*m[4,3,2,1]+2976*m [5 ,1,1,1]-11264*m[3,3,2,1]+18176*m[4,2,2,1]-4096*m[4,4,2,2]+18432*m[3,3,3,1]-7 68 *m[5,3,1]-256*m[6,2,2]+384*m[5,2,1]-3072*m[3,3,3]+1024*m[4,4,1]+2112*m[3,2,2 ]- 1024*m[4,4,2]-128*m[4,3,1]+768*m[5,2,2]-16*m[3,3]-344*m[4,1,1]-288*m[5,2]-36 *m [4,2]-8832*m[2,2,2,1]-2016*m[3,3,1])*SQRs5^2-240*m[5,3,1]+384*m[5,3,2]-256*m [5 ,3,3]-96*m[6,2,2]+72*m[5,2,1]+96*m[3,3,3]+320*m[4,4,1]-576*m[4,4,2]-384*m[4, 3, 3]-16*m[7,1,1]+(-256*m[2,1,1]+144*m[1,1,1]+96*m[2,1]-256*m[2,2,2]-40*m[1,1]+ 256*m[2,2,1]-96*m[2,2]+224*m[1,1,1,1]-28*m[2])*SQRs5^6-256*m[4,4,4]-40*m[4,3 ,1 ]-288*m[5,2,2]-SQRs5^8; > Given values a1, a2, a3, a4, a5, > I am searching for a way > to express the product of all terms of the form sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), as a polynomial in the values sin(ai)^2, i=1,..,5, > where the variables eps1, eps2, eps3, eps4, eps5 > take on values (+1) and (-1), resp., > but not more than half of these five values are (-1). As an example, for three terms instead of five, > this would mean the following computation: > sin(a+b+c)*sin(-a+b+c)*sin(a-b+c)*sin(a+b-c); > expand(eval(expand(%),cos=(x->sqrt(1-sin(x)^2)))); -sin(a + b + c) sin(a - b - c) sin(a - b + c) sin(a + b - c) 2 2 2 4 4 4 > -4 sin(a) sin(b) sin(c) - sin(a) - sin(b) - sin(c) 2 2 2 2 2 2 > + 2 sin(a) sin(b) + 2 sin(a) sin(c) + 2 sin(b) sin(c) > collect(numer(eval(%,sin=(x->x/r))),r,factor); 2 2 2 2 > -(a + b - c) (a + b + c) (a - b + c) (a - b - c) r - 4 a b c The problem is that there are 16 sine factors, > each of which expands into a sum of 2^5 products > of sines and cosines of ai; so if I expand the whole product, > there will be (2^5)^16 = 2^80 > 10^8 terms, which is quite a lot. > (I was hoping to do the same calculation > for 7 or 9 variables ai, or even more... ;-( ) For people who are interested where these expressions come from: > According to A. F. Moebius, > Ueber die Gleichungen, mittelst welcher > aus den Seiten eines in einen Kreis zu beschreibenden Vielecks > der Halbmesser des Kreises und die Flache > des Vielecks gefunden werden > [Crelle's Journal 1828 Band 3 p. 5-34], > Gesammelte Werke, vol 1., pp. 407-438. http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a 1 da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=ima g e if we substitute sin(ai) := ai/r, i=1,2,... into the expression above, we obtain a polynomial in r > whose largest root corresponds to the circumcenter radius > of the convex cyclic polynomial with side lengths a1,a2,.... Also, there are similar formulas with which one can > solve the following circle packing problem: Given a cyclic sequence of circles, find the radius of a circle > around which the given circles can be arranged so that > each of them touches its two neighbours in the sequence > and the central circle. Incidentally, as a side product of the paper cited above, > Moebius proves Robbins' Conjecture about the degree > of generalized Heron polynomials of cyclic polygons > (D. P. Robbins, Areas of Polygons Inscribed in a Circle, > Discrete & Computational Geometry 12, 223-236, 1994). === Subject: Re: To simplify a product of 16 sines in 5 variables or better, let m[n1,n2,...] denote sum of all monomials in SQRs1,SQRs2,SQRs3,SQRs4, SQRs5 with degree n1,n2,... then : ANS:=-384*m[4,3,3]-5152*m[4,2,2,2]-3072*m[3,3,3,2]-36*m[4,2,2]+736*m[5,3,1,1 ]+4096*m[4,4,2,2,1]+208*m[4,3,1,1]+32*m[7,1,1,1]-1984*m[5,3,1,1,1]-16000*m[3 , 3,3,1,1]+56*m[5,3]+384*m[5,3,2]+1520*m[2,2,2,1,1]-2304*m[5,2,2,1,1]-2880*m[5 , 1,1,1,1]-12288*m[3,3,3,3,1]-4096*m[4,4,2,2,2]+512*m[4,4,3]-8192*m[4,3,2,2,1] - 576*m[4,4,2]+416*m[3,3,1,1]+192*m[4,3,2]-8832*m[2,2,2,2,1]+752*m[4,1,1,1,1]- 4 096*m[4,4,2,1,1]+2112*m[3,2,2,2]-38400*m[3,2,2,2,2]-2016*m[3,3,2,1]+256*m[6, 2 ,2,1]+512*m[5,3,3,1]+4352*m[4,3,2,1,1]+768*m[5,2,2,2]-m[8]-768*m[5,3,2,1]-25 6 *m[6,2,2,2]+18176*m[4,2,2,2,1]+5056*m[3,3,2,1,1]-344*m[4,2,1,1]+72*m[5,2,1]- 4 0*m[6,1,1]+272*m[3,2,2,1]+1536*m[5,3,2,1,1]-1280*m[4,3,3,1]+176*m[5,1,1,1]+2 5 60*m[4,3,3,1,1]+2048*m[3,3,3,3]+144*m[6,1,1,1]+2048*m[4,3,2,2]-256*m[6,2,1,1 ] +5120*m[3,2,2,2,1]+38912*m[3,3,2,2,2]-16384*m[3,3,3,3,3]-70*m[4,4]+1024*m[4, 4 ,2,1]-16*m[7,1,1]+18432*m[3,3,3,2,1]-1024*m[4,4,1,1]-11264*m[3,3,2,2,1]-8320 * m[4,2,2,1,1]-256*m[5,3,3]+224*m[6,1,1,1,1]+48768*m[2,2,2,2,2]-256*m[4,4,4]-2 2 272*m[4,2,2,2,2]+4928*m[3,3,3,1]-40*m[4,3,1]-336*m[5,2,1,1]+384*m[5,2,2,1]+5 1 2*m[3,3,2,2]-1024*m[4,4,2,2]-240*m[5,3,1]-96*m[6,2,2]-36864*m[3,3,3,2,2]+124 8 *m[4,2,2,1]+2976*m[5,2,1,1,1]+8*m[7,1]-128*m[4,3,2,1]+96*m[6,2,1]-160*m[3,2, 2 ,1,1]-928*m[3,2,1,1,1]+384*m[3,3,1,1,1]-288*m[5,2,2]-28*m[6,2]-64*m[7,1,1,1, 1 ]-2008*m[2,2,2,2]-3232*m[4,3,1,1,1]-16*m[3,3,2]+24576*m[3,3,3,3,2]-1024*m[5, 3 ,3,1,1]+320*m[4,4,1]+2544*m[4,2,1,1,1]+4096*m[4,4,1,1,1]+96*m[3,3,3]+8192*m[ 4 ,3,2,2,2]; > Hi ! Let SQRs.i denote sin(a.i)^2, let m[n1,n2,...] denote sum of all monomials in SQRs1,SQRs2,SQRs3,SQRs4, (not SQRs5) with degree n1,n2,... then the product can be > expressed as ANS. ANS := (-2016*m[3,3,2]-256*m[6,2,1]-128*m[4,3,2]-40*m[6,1]-344*m[4,2,1]+176*m[ > 5,1,1]-928*m[3,2,1,1]+752*m[4,1,1,1]-1984*m[5,3,1,1]+18176*m[4,2,2,2]-240*m[ 5 , > 3]-16*m[7,1]+96*m[6,2]+144*m[6,1,1]+1248*m[4,2,2]+320*m[4,4]-40*m[4,3]-160*m [ 3 > ,2,2,1]-3232*m[4,3,1,1]+2976*m[5,2,1,1]-64*m[7,1,1,1]-12288*m[3,3,3,3]+1536* m [ > 5,3,2,1]-4096*m[4,4,2,1]+2560*m[4,3,3,1]-1024*m[5,3,3,1]+5120*m[3,2,2,2]-883 2 * > m[2,2,2,2]-8192*m[4,3,2,2]-11264*m[3,3,2,2]+18432*m[3,3,3,2]+2544*m[4,2,1,1] + > 384*m[3,3,1,1]+4096*m[4,4,1,1]+4352*m[4,3,2,1]-2880*m[5,1,1,1]-2304*m[5,2,2, 1 ] > +224*m[6,1,1,1]+5056*m[3,3,2,1]-8320*m[4,2,2,1]+4096*m[4,4,2,2]-16000*m[3,3, 3 , > 1]+736*m[5,3,1]-768*m[5,3,2]+512*m[5,3,3]+256*m[6,2,2]-336*m[5,2,1]+4928*m[3 , 3 > ,3]-1024*m[4,4,1]+272*m[3,2,2]+1024*m[4,4,2]-1280*m[4,3,3]+32*m[7,1,1]+208*m [ 4 > ,3,1]+384*m[5,2,2]+72*m[5,2]+1520*m[2,2,2,1]+8*m[7]+416*m[3,3,1])*SQRs5-16*m [ 3 > ,3,2]+96*m[6,2,1]+192*m[4,3,2]+(-3072*m[3,3,2]-128*m[4,2,1]+736*m[5,1,1]+384 * m > [3,1,1,1]+5056*m[3,2,1,1]-3232*m[4,1,1,1]-1024*m[5,3,1,1]+8192*m[4,2,2,2]-25 6 * > m[5,3]-160*m[2,2,1,1]+2048*m[4,2,2]+512*m[4,4]-384*m[4,3]+416*m[3,1,1]-40*m[ 4 , > 1]-11264*m[3,2,2,1]+2560*m[4,3,1,1]+1536*m[5,2,1,1]-2016*m[3,2,1]-16384*m[3, 3 , > 3,3]-240*m[5,1]+38912*m[3,2,2,2]-38400*m[2,2,2,2]-928*m[2,1,1,1]-36864*m[3,3 , 2 > ,2]+24576*m[3,3,3,2]+4352*m[4,2,1,1]-16000*m[3,3,1,1]+2112*m[2,2,2]-1984*m[5 , 1 > ,1,1]+18432*m[3,3,2,1]-8192*m[4,2,2,1]-12288*m[3,3,3,1]+512*m[5,3,1]-768*m[5 , 2 > ,1]+2048*m[3,3,3]+512*m[3,2,2]+272*m[2,2,1]-1280*m[4,3,1]+96*m[3,3]+208*m[4, 1 , > 1]+384*m[5,2]-16*m[3,2]+192*m[4,2]+5120*m[2,2,2,1]+56*m[5]+4928*m[3,3,1])* > SQRs5^3+(1536*m[3,2,1,1]-2304*m[2,2,1,1]+736*m[3,1,1]-768*m[3,2,1]-336*m[2,1 , 1 > ]+176*m[1,1,1]+2976*m[2,1,1,1]+72*m[2,1]+768*m[2,2,2]-1984*m[3,1,1,1]+384*m[ 2 , > 2,1]-240*m[3,1]-288*m[2,2]-256*m[3,3]+384*m[3,2]+512*m[3,3,1]-2880*m[1,1,1,1 ] + > 56*m[3]-1024*m[3,3,1,1])*SQRs5^5+736*m[5,3,1,1]-5152*m[4,2,2,2]+56*m[5,3]+8* m [ > 7,1]-28*m[6,2]+(1024*m[4,2,1]-3232*m[3,1,1,1]+4352*m[3,2,1,1]+4096*m[4,1,1,1 ] - > 4096*m[4,2,2,2]-8320*m[2,2,1,1]-1024*m[4,2,2]-256*m[4,4]+512*m[4,3]+208*m[3, 1 , > 1]+320*m[4,1]-70*m[4]-8192*m[3,2,2,1]+752*m[1,1,1,1]-128*m[3,2,1]-344*m[2,1, 1 ] > +8192*m[3,2,2,2]-22272*m[2,2,2,2]+2544*m[2,1,1,1]-4096*m[4,2,1,1]+2560*m[3,3 , 1 > ,1]-5152*m[2,2,2]+4096*m[4,2,2,1]+2048*m[3,2,2]+1248*m[2,2,1]-40*m[3,1]-36*m [ 2 > ,2]-384*m[3,3]-1024*m[4,1,1]+192*m[3,2]-576*m[4,2]+18176*m[2,2,2,1]-1280*m[3 , 3 > ,1])*SQRs5^4-40*m[6,1,1]-36*m[4,2,2]+(8*m[1]-16*m[1,1]+32*m[1,1,1]-64*m[1,1, 1 , > 1])*SQRs5^7+512*m[4,4,3]-70*m[4,4]+272*m[3,2,2,1]+208*m[4,3,1,1]-336*m[5,2,1 , 1 > ]+32*m[7,1,1,1]-256*m[6,2,1,1]+768*m[5,2,2,2]+2048*m[3,3,3,3]+256*m[6,2,2,1] - > 768*m[5,3,2,1]+1024*m[4,4,2,1]-1280*m[4,3,3,1]+512*m[5,3,3,1]+2112*m[3,2,2,2 ] - > 2008*m[2,2,2,2]+2048*m[4,3,2,2]+512*m[3,3,2,2]-3072*m[3,3,3,2]-344*m[4,2,1,1 ] + > 416*m[3,3,1,1]-1024*m[4,4,1,1]-m[8]-128*m[4,3,2,1]+176*m[5,1,1,1]+384*m[5,2, 2 , > 1]+144*m[6,1,1,1]-2016*m[3,3,2,1]+1248*m[4,2,2,1]-1024*m[4,4,2,2]-256*m[6,2, 2 , > 2]+4928*m[3,3,3,1]+(512*m[3,3,2]+256*m[6,2,1]+2048*m[4,3,2]+96*m[6,1]+1248*m [ 4 > ,2,1]-336*m[5,1,1]-928*m[3,1,1,1]-160*m[3,2,1,1]+2544*m[4,1,1,1]+1536*m[5,3, 1 , > 1]-22272*m[4,2,2,2]+384*m[5,3]-96*m[6,2]+1520*m[2,2,1,1]-256*m[6,1,1]-5152*m [ 4 > ,2,2]-576*m[4,4]+192*m[4,3]+5120*m[3,2,2,1]+4352*m[4,3,1,1]-2304*m[5,2,1,1]+ > 272*m[3,2,1]+24576*m[3,3,3,3]+4096*m[4,4,2,1]+72*m[5,1]-28*m[6]-38400*m[3,2, 2 , > 2]+48768*m[2,2,2,2]+8192*m[4,3,2,2]+38912*m[3,3,2,2]-36864*m[3,3,3,2]-8320*m [ 4 > ,2,1,1]+5056*m[3,3,1,1]-4096*m[4,4,1,1]-2008*m[2,2,2]-8192*m[4,3,2,1]+2976*m [ 5 > ,1,1,1]-11264*m[3,3,2,1]+18176*m[4,2,2,1]-4096*m[4,4,2,2]+18432*m[3,3,3,1]-7 6 8 > *m[5,3,1]-256*m[6,2,2]+384*m[5,2,1]-3072*m[3,3,3]+1024*m[4,4,1]+2112*m[3,2,2 ] - > 1024*m[4,4,2]-128*m[4,3,1]+768*m[5,2,2]-16*m[3,3]-344*m[4,1,1]-288*m[5,2]-36 * m > [4,2]-8832*m[2,2,2,1]-2016*m[3,3,1])*SQRs5^2-240*m[5,3,1]+384*m[5,3,2]-256*m [ 5 > ,3,3]-96*m[6,2,2]+72*m[5,2,1]+96*m[3,3,3]+320*m[4,4,1]-576*m[4,4,2]-384*m[4, 3 , > 3]-16*m[7,1,1]+(-256*m[2,1,1]+144*m[1,1,1]+96*m[2,1]-256*m[2,2,2]-40*m[1,1]+ > 256*m[2,2,1]-96*m[2,2]+224*m[1,1,1,1]-28*m[2])*SQRs5^6-256*m[4,4,4]-40*m[4,3 , 1 > ]-288*m[5,2,2]-SQRs5^8; > Given values a1, a2, a3, a4, a5, > I am searching for a way > to express the product of all terms of the form > sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), > as a polynomial in the values sin(ai)^2, i=1,..,5, > where the variables eps1, eps2, eps3, eps4, eps5 > take on values (+1) and (-1), resp., > but not more than half of these five values are (-1). > As an example, for three terms instead of five, > this would mean the following computation: > sin(a+b+c)*sin(-a+b+c)*sin(a-b+c)*sin(a+b-c); >expand(eval(expand(%),cos=(x->sqrt(1-sin(x)^2)))); > -sin(a + b + c) sin(a - b - c) sin(a - b + c) sin(a + b - c) > 2 2 2 4 4 4 > -4 sin(a) sin(b) sin(c) - sin(a) - sin(b) - sin(c) > 2 2 2 2 2 2 > + 2 sin(a) sin(b) + 2 sin(a) sin(c) + 2 sin(b) sin(c) > collect(numer(eval(%,sin=(x->x/r))),r,factor); > 2 2 2 2 > -(a + b - c) (a + b + c) (a - b + c) (a - b - c) r - 4 a b c > The problem is that there are 16 sine factors, > each of which expands into a sum of 2^5 products > of sines and cosines of ai; so if I expand the whole product, > there will be > (2^5)^16 = 2^80 > 10^8 > terms, which is quite a lot. > (I was hoping to do the same calculation > for 7 or 9 variables ai, or even more... ;-( ) > For people who are interested where these expressions come from: > According to > A. F. Moebius, > Ueber die Gleichungen, mittelst welcher > aus den Seiten eines in einen Kreis zu beschreibenden Vielecks > der Halbmesser des Kreises und die Flache > des Vielecks gefunden werden > [Crelle's Journal 1828 Band 3 p. 5-34], > Gesammelte Werke, vol 1., pp. 407-438. > http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a 1 da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=ima g e > if we substitute > sin(ai) := ai/r, i=1,2,... > into the expression above, we obtain a polynomial in r > whose largest root corresponds to the circumcenter radius > of the convex cyclic polynomial with side lengths a1,a2,.... > Also, there are similar formulas with which one can > solve the following circle packing problem: > Given a cyclic sequence of circles, find the radius of a circle > around which the given circles can be arranged so that > each of them touches its two neighbours in the sequence > and the central circle. > Incidentally, as a side product of the paper cited above, > Moebius proves Robbins' Conjecture about the degree > of generalized Heron polynomials of cyclic polygons > (D. P. Robbins, Areas of Polygons Inscribed in a Circle, > Discrete & Computational Geometry 12, 223-236, 1994). === Subject: Re: To simplify a product of 16 sines in 5 variables >>Given values a1, a2, a3, a4, a5, >>I am searching for a way >>to express the product of all terms of the form >> sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), >>as a polynomial in the values sin(ai)^2, i=1,..,5, >>where the variables eps1, eps2, eps3, eps4, eps5 >>take on values (+1) and (-1), resp., >>but not more than half of these five values are (-1). > This Maple code obtains such an expression, I only read your message after I posted my reply to the newsgroups. > but it doesn't appear useful. Maybe someone will be able to spot relations between the exponents of the sines, and their coefficients, and will be able to find some deeper explanation/meaning. (Maybe not... ;-I ) Anyway, as far as I know, you might be the first person to have come up with this *explicit* formula. (Unfortunately, I don't know much. ) There have been other people around who published similar formulas for the *area* of a cyclic pentagon in terms of its side lengths. (I also have a similar formula for the product of circumscribed radius and area of a cyclic pentagon, for which I suppose also nobody cares... :-I ) Maybe this formula here also has the form of a discriminant (whatever this means - I will have to check with Robbin's paper), maybe not... > The program takes about two minutes to run. It seems to run faster than the code I posted. But I will have to wait till tomorrow for an in-detail-inspection, and also for testing whether your code gets results for higher, odd values of n. > pmone := [-1, +1]; > # Take the product >p1:= 1; > for eps1 in pmone do > for eps2 in pmone do > for eps3 in pmone do > for eps4 in pmone do > for eps5 in pmone do > if (eps1 + eps2 + eps3 + eps4 + eps5 > 0) then > p1 := p1* sin(a1*eps1 + a2*eps2 + a3*eps3 + a4*eps4 + a5*eps5); > fi; > od;od;od;od;od; > p1; > ># Change angles to 2*a1 through 2*a5. > # Look for a formuls in cos(2*a1) through cos(2*a5); >a1 := a1twice/2; > a2 := a2twice/2; > a3 := a3twice/2; > a4 := a4twice/2; > a5 := a5twice/2; >p2 := combine(p1); # Sums of cosine(sum of angles) > p3 := expand(p2); # Go back to 2*a1 through 2*a5 >p4 := expand(subs( > cos(a1twice) = 1 - 2*sqsa1, > cos(a2twice) = 1 - 2*sqsa2, > cos(a3twice) = 1 - 2*sqsa3, > cos(a4twice) = 1 - 2*sqsa4, > cos(a5twice) = 1 - 2*sqsa5, p3)); > # Substitute cosine values in terms of sin^2 Once again, thank you very much! Thomas === Subject: Re: To simplify a product of 16 sines in 5 variables > Given values a1, a2, a3, a4, a5, > I am searching for a way > to express the product of all terms of the form > sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), >as a polynomial in the values sin(ai)^2, i=1,..,5, > where the variables eps1, eps2, eps3, eps4, eps5 > take on values (+1) and (-1), resp., > but not more than half of these five values are (-1). As usual, once you ask a question, you become more likely to find the answer yourself. But first, I would like to thank David M Einstein and CW for their replies by Email. The suggestion David made for a simplification did not really help in practice, because Maple still did not find a result within 45 minutes, before I stopped the experiment. I found it also interesting that only expanding the eps5*a5-summands in all sine terms in the product above yields a result which also contains cos(a5) terms of _odd_ order. The coefficients of these must add up to zero, but I found it hard to let Maple manage these terms automatically; and the compexity of the coefficients of the other terms in a5 got high (and somehow made my compute crash), so I gave up on this route. Chris asked for the actual expression I want to simplify. - In Maple notation these are: mul( sin(add(j[i]*a||i,i=1..n)), j = select(x->add(i,i=x)>=0,combinat[permute]([seq]([1,-1][],i=1..n),n)) ); for odd values n = 5, 7, 9, ... E.g., for n:= 5, I wanted to express - sin(a1-a2+a3-a4+a5)*sin(a1-a2+a3+a4-a5)*sin(a1-a2+a3+a4+a5) *sin(a1-a2-a3+a4+a5)*sin(a1+a2-a3-a4+a5)*sin(a1+a2-a3+a4-a5) *sin(a1+a2-a3+a4+a5)*sin(a1+a2+a3-a4-a5)*sin(a1+a2+a3-a4+a5) *sin(a1+a2+a3+a4-a5)*sin(a1+a2+a3+a4+a5)*sin(a1-a2-a3+a4-a5) *sin(a1-a2-a3-a4+a5)*sin(a1-a2-a3-a4-a5)*sin(a1-a2+a3-a4-a5) *sin(a1+a2-a3-a4-a5) in terms of sin(ai)^2, i=1..5. > The problem is that there are 16 sine factors, > each of which expands into a sum of 2^5 products > of sines and cosines of ai; so if I expand the whole product, > there will be > (2^5)^16 = 2^80 > 10^24 >terms, which is quite a lot. > ;-( ) > According to > A. F. Moebius, > Ueber die Gleichungen, mittelst welcher > aus den Seiten eines in einen Kreis zu beschreibenden Vielecks > der Halbmesser des Kreises und die Flache > des Vielecks gefunden werden > [Crelle's Journal 1828 Band 3 p. 5-34], > Gesammelte Werke, vol 1., pp. 407-438. > http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a 1 da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=ima g e >if we substitute > sin(ai) := ai/r, i=1,2,... >into the expression above, we obtain a polynomial in r > whose largest root corresponds to the circumcenter radius > of the convex cyclic polynomial with side lengths a1,a2,.... The following Maple code got me (one of) the result(s) I was looking for: n := 5; select(x->add(i,i=x)>=0,combinat[permute]([seq]([1,-1][],i=1..n),n)); mul(sin(add(j[i]*a||i,i=1..n)),j=%); expand(eval(expand(combine(%,trig),sin),cos=(x->sqrt(1-sin(x)^2)))); collect(numer(eval(%,sin=(x->x/r))),r,expand); The trick here was to use the combine,trig command in order to turn the product of sines into a sum of trigonometric terms. The resulting formula for the radius r of an inscribed pentagon with side lengths a1,a2,...,a5 was: (Sym(72*a4^2*a3^10*a2^4)+Sym(56*a4^10*a1^6) +Sym(-928*a2^6*a4^2*a5^2*a1^4*a3^2)+Sym(176*a1^10*a2^2*a4^2*a5^2) +Sym(-16*a4^6*a3^4*a2^6)+Sym(-40*a3^2*a5^12*a1^2) +Sym(-36*a3^8*a2^4*a1^4)+Sym(-28*a1^12*a2^4) +Sym(752*a4^8*a1^2*a3^2*a5^2*a2^2)+Sym(-70*a1^8*a4^8) +Sym(272*a4^6*a3^4*a1^4*a5^2)+Sym(-344*a4^8*a3^4*a1^2*a5^2) +Sym(8*a1^14*a4^2)+Sym(416*a3^2*a4^6*a1^6*a5^2) +Sym(1520*a3^2*a4^4*a1^4*a5^4*a2^2)+Sym(-40*a1^8*a2^6*a5^2) +Sym(-2008*a1^4*a2^4*a4^4*a3^4)+Sym(-a1^16)) * r^14 +(Sym(1248*a1^4*a2^8*a4^4*a5^2)+Sym(-336*a1^2*a2^10*a4^2*a5^4) +Sym(192*a2^8*a3^6*a4^4)+Sym(208*a4^6*a3^2*a2^8*a5^2) +Sym(96*a1^6*a2^6*a3^6)+Sym(-160*a1^4*a2^6*a4^2*a5^2*a3^4) +Sym(96*a3^4*a5^12*a1^2)+Sym(2544*a1^4*a2^8*a4^2*a5^2*a3^2) +Sym(-2016*a2^2*a4^6*a3^4*a5^6)+Sym(320*a1^2*a3^8*a2^8) +Sym(-288*a4^10*a2^4*a5^4)+Sym(-240*a4^10*a1^2*a2^6) +Sym(2112*a4^6*a3^4*a2^4*a5^4)+Sym(-2880*a1^2*a4^2*a5^10*a2^2*a3^2) +Sym(-8832*a1^4*a2^4*a4^2*a3^4*a5^4)+Sym(384*a1^2*a2^6*a3^6*a4^2*a5^2) +Sym(144*a4^12*a3^2*a1^2*a5^2)+Sym(-16*a5^14*a3^2*a1^2)) * r^12 +(Sym(-128*a1^6*a2^8*a4^4*a5^2)+Sym(-8320*a1^4*a2^8*a4^4*a5^2*a3^2) +Sym(384*a2^10*a4^4*a1^2*a5^4)+Sym(-576*a3^8*a2^8*a4^4) +Sym(5056*a1^6*a2^6*a4^2*a5^2*a3^4)+Sym(-3232*a1^6*a2^8*a4^2*a5^2*a3^2) +Sym(5120*a1^2*a2^4*a3^4*a4^4*a5^6)+Sym(2976*a1^2*a2^10*a3^2*a4^2*a5^4) +Sym(4928*a2^2*a4^6*a3^6*a5^6)+Sym(512*a4^6*a3^4*a2^4*a5^6) +Sym(384*a4^10*a2^4*a5^6)+Sym(736*a1^10*a2^2*a4^6*a5^2) +Sym(-5152*a1^4*a4^4*a3^4*a5^8)+Sym(-96*a3^4*a5^12*a1^4) +Sym(-384*a4^8*a2^6*a5^6)+Sym(48768*a1^4*a2^4*a4^4*a3^4*a5^4) +Sym(-1024*a2^2*a1^8*a4^8*a5^2)+Sym(224*a4^12*a3^2*a1^2*a5^2*a2^2) +Sym(-256*a2^4*a3^2*a5^12*a1^2)+Sym(32*a2^2*a3^14*a1^2*a5^2)) * r^10 +(Sym(4352*a1^6*a2^8*a4^4*a5^2*a3^2)+Sym(18176*a1^4*a2^8*a4^4*a5^2*a3^4) +Sym(-768*a2^10*a4^6*a1^2*a5^4)+Sym(4096*a2^8*a1^8*a4^2*a5^2*a3^2) +Sym(-38400*a1^4*a2^6*a4^4*a5^4*a3^4)+Sym(-1280*a4^6*a2^8*a1^2*a5^6) +Sym(-1984*a1^2*a2^10*a3^2*a4^2*a5^6) +Sym(-2304*a1^10*a2^2*a3^2*a4^4*a5^4) +Sym(-11264*a1^6*a2^6*a4^4*a5^2*a3^4)+Sym(-256*a4^6*a3^10*a2^6) +Sym(-16000*a1^2*a4^6*a3^6*a2^2*a5^6)+Sym(-3072*a4^6*a3^6*a2^4*a5^6) +Sym(1024*a4^8*a2^4*a1^8*a5^2)+Sym(2048*a2^4*a1^8*a4^6*a5^4) +Sym(-64*a4^14*a3^2*a1^2*a5^2*a2^2)+Sym(256*a2^4*a3^4*a5^12*a1^2) +Sym(768*a2^4*a3^4*a5^10*a1^4)+Sym(512*a5^8*a3^8*a2^6)) * r^8 +(Sym(-1024*a2^8*a1^8*a4^4*a5^4)+Sym(-8192*a1^6*a2^8*a4^4*a5^2*a3^4) +Sym(-4096*a2^8*a1^8*a4^4*a5^2*a3^2)+Sym(-22272*a2^4*a1^8*a4^4*a5^4*a3^4) +Sym(38912*a1^6*a2^6*a4^4*a5^4*a3^4)+Sym(512*a2^10*a3^2*a4^6*a5^6) +Sym(2560*a4^6*a3^2*a2^8*a1^2*a5^6)+Sym(1536*a2^10*a4^4*a3^2*a1^2*a5^6) +Sym(18432*a1^2*a4^6*a3^6*a2^4*a5^6)+Sym(2048*a1^6*a4^6*a2^6*a5^6) +Sym(-256*a2^4*a3^4*a5^12*a1^4)+Sym(-256*a5^8*a3^8*a2^8)) * r^6 +(Sym(8192*a4^6*a2^8*a1^4*a3^4*a5^4)+Sym(-1024*a2^10*a3^2*a4^6*a1^2*a5^6) +Sym(4096*a2^8*a1^8*a4^4*a3^2*a5^4)+Sym(-12288*a1^6*a4^6*a2^6*a3^2*a5^6) +Sym(-36864*a1^4*a2^6*a3^6*a4^4*a5^6)) * r^4 +(Sym(-4096*a2^4*a1^8*a3^4*a4^8*a5^4) +Sym(24576*a1^6*a2^6*a3^6*a4^4*a5^6)) * r^2 + Sym(-16384*a1^6*a2^6*a3^6*a4^6*a5^6) = 0; where Sym stands short for symmetrization, e.g., Sym(-a1^16) = - a1^16 - a2^16 - a3^16 - a4^16 - a5^16, Sym(-16384*a1^6*a2^6*a3^6*a4^6*a5^6) = -16384*a1^6*a2^6*a3^6*a4^6*a5^6. Maybe someone will be able to spot a relation between the exponents of the side lenghts ai and the signs and coefficients in this formula, which might be generalized to higher values of n ?! === Subject: Re: To simplify a product of 16 sines in 5 variables > Given values a1, a2, a3, a4, a5, >I am searching for a way >to express the product of all terms of the form > sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), > as a polynomial in the values sin(ai)^2, i=1,..,5, >where the variables eps1, eps2, eps3, eps4, eps5 >take on values (+1) and (-1), resp., >but not more than half of these five values are (-1). > -> As usual, once you ask a question, you become more likely > to find the answer yourself. > But first, I would like to thank David M Einstein and CW > for their replies by Email. > The suggestion David made for a simplification > did not really help in practice, because Maple still did not > find a result within 45 minutes, before I stopped the experiment. > I found it also interesting that only expanding the eps5*a5-summands > in all sine terms in the product above > yields a result which also contains cos(a5) terms of _odd_ order. > The coefficients of these must add up to zero, > but I found it hard to let Maple manage these terms automatically; > and the compexity of the coefficients of the other terms in a5 > got high (and somehow made my compute crash), > so I gave up on this route. > Chris asked for the actual expression I want to simplify. - > In Maple notation these are: > mul( > sin(add(j[i]*a||i,i=1..n)), > j = select(x->add(i,i=x)>=0,combinat[permute]([seq]([1,-1][],i=1..n),n)) > ); > for odd values n = 5, 7, 9, ... > E.g., for n:= 5, I wanted to express > - sin(a1-a2+a3-a4+a5)*sin(a1-a2+a3+a4-a5)*sin(a1-a2+a3+a4+a5) > *sin(a1-a2-a3+a4+a5)*sin(a1+a2-a3-a4+a5)*sin(a1+a2-a3+a4-a5) > *sin(a1+a2-a3+a4+a5)*sin(a1+a2+a3-a4-a5)*sin(a1+a2+a3-a4+a5) > *sin(a1+a2+a3+a4-a5)*sin(a1+a2+a3+a4+a5)*sin(a1-a2-a3+a4-a5) > *sin(a1-a2-a3-a4+a5)*sin(a1-a2-a3-a4-a5)*sin(a1-a2+a3-a4-a5) > *sin(a1+a2-a3-a4-a5) I still think that this approach has promise. Rewrite the product as - sin(a1 + a2 + a3 + a4 + a5) * sin(a1 + a2 + a3 + a4 - a5) > * sin(a1 + a2 + a3 - a4 + a5) * sin(a1 + a2 + a3 - a4 - a5) > * sin(a1 + a2 - a3 + a4 + a5) * sin(a1 + a2 - a3 + a4 - a5) > * sin(a1 + a2 - a3 - a4 + a5) * sin(a1 + a2 - a3 - a4 - a5) > * sin(a1 - a2 + a3 + a4 + a5) * sin(a1 - a2 + a3 + a4 - a5) > * sin(a1 - a2 + a3 - a4 + a5) * sin(a1 - a2 + a3 - a4 - a5) > * sin(- a1 + a2 + a3 + a4 + a5) * sin(- a1 + a2 + a3 + a4 - a5) > * sin(-a1 + a2 + a3 - a4 + a5) * sin(-a1 + a2 + a3 - a4 - a5) Now apply the identity sin(a+b)sin(a-b)=sin^2 a - sin^2 b > (1) to each row to get the product -(sin^2 ( a1 + a2 + a3 + a4) - sin^2 a5) * (sin^2 ( a1 + a2 + a3 - a4) > - sin^2 a5) > *(sin^2 ( a1 + a2 - a3 + a4) - sin^2 a5) * (sin^2 ( a1 + a2 - a3 - a4) > - sin^2 a5) > *(sin^2 ( a1 - a2 + a3 + a4) - sin^2 a5) *(sin^2 ( a1 - a2 + a3 - a4) - > sin^2 a5) > *(sin^2 ( -a1 + a2 + a3 + a4) - sin^2 a5) * (sin^2 ( -a1 + a2 + a3 - > a4) - sin^2 a5) Now, we can apply the identity (1) along with the identity sin^2 (a + b) + sin^2 ( a - b ) = 2 sin^2 a - 4 sin^2 a sin^2 b + 2 > sin^2 b to the product of each of the above rows to get -(sin^4 ( a1 + a2 + a3) + p1 sin^2 ( a1 + a2 + a3) + p2) > *(sin^4 ( a1 + a2 - a3) + p1 sin^2 ( a1 + a2 - a3) + p2) > *(sin^4 ( a1 - a2 + a3) + p1 sin^2 ( a1 - a2 + a3) + p2) > *(sin^4 ( a1 - a2 + a3) - p1 sin^2 ( a1 - a2 - a3) + p2) That last line should be *(sin^4 ( a1 - a2 - a3) - p1 sin^2 ( a1 - a2 - a3) + p2) of course. where p1 and p2 are polynomials in sin^2 a4 and sin^2 a5 Now add in the identity for sin^4 (a + b) + sin^4 (a - b) = (sin^2 (a + b) + sin^2 (a-b) )^2 - 2 sin^2 (a + b) sin^2 (a-b) = sin^4 (a + b) + sin^4 (a - b) = 16 sin^4 a sin^4 b - 16 sin^4 a sin^2 b - 16 sin^2 a sin^4 b + 2 sin^4 a + 12 sin^2 a sin^2 b + 2 sin^4 b and reduce the product of consecutive terms obtaining a product of the form -(sin^8 (a1 + a2) + p3 sin^6 (a1 + a2) + p4 sin^4 (a1+a2) + p5 sin^2 (a1+ a2) + p6) *(sin^8 (a1 - a2) + p3 sin^6 (a1 - a2) + p4 sin^4 (a1-a2) + p5 sin^2 (a1- a2) + p6) where p3, p4, p5, and p6 are polynomials in sin^2 a3, sin^2 a4, and sin^2 a5 finally use all the above identities along with sin^6 (a + b) + sin^6 (a - b) = > sin^8 (a + b) + sin^8 (a - b) = I am still too lazy to expand these, but this should be simple for a computer. to get the final result. There are probably still sign and/or sine errors in the above. I don't have a working cas, but this should be able to get the results > fairly quickly, and may illuminate the structure or the result. I still do not see how this helps us advance towards greater understanding, but I think that it is somewhat cute. > > in terms of sin(ai)^2, i=1..5. > > The problem is that there are 16 sine factors, >each of which expands into a sum of 2^5 products >of sines and cosines of ai; so if I expand the whole product, >there will be > (2^5)^16 = 2^80 > 10^24 > terms, which is quite a lot. > ;-( ) > According to > A. F. Moebius, > Ueber die Gleichungen, mittelst welcher > aus den Seiten eines in einen Kreis zu beschreibenden Vielecks > der Halbmesser des Kreises und die Flache > des Vielecks gefunden werden > [Crelle's Journal 1828 Band 3 p. 5-34], > Gesammelte Werke, vol 1., pp. 407-438. > http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a 1 da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=ima g e > if we substitute > sin(ai) := ai/r, i=1,2,... > into the expression above, we obtain a polynomial in r >whose largest root corresponds to the circumcenter radius >of the convex cyclic polynomial with side lengths a1,a2,.... > The following Maple code got me (one of) the result(s) > I was looking for: > > n := 5; > select(x->add(i,i=x)>=0,combinat[permute]([seq]([1,-1][],i=1..n),n)); > mul(sin(add(j[i]*a||i,i=1..n)),j=%); > expand(eval(expand(combine(%,trig),sin),cos=(x->sqrt(1-sin(x)^2)))); > collect(numer(eval(%,sin=(x->x/r))),r,expand); > > The trick here was to use the combine,trig command > in order to turn the product of sines into > a sum of trigonometric terms. > > The resulting formula for the radius r of an inscribed pentagon > with side lengths a1,a2,...,a5 was: > > (Sym(72*a4^2*a3^10*a2^4)+Sym(56*a4^10*a1^6) > +Sym(-928*a2^6*a4^2*a5^2*a1^4*a3^2)+Sym(176*a1^10*a2^2*a4^2*a5^2) > +Sym(-16*a4^6*a3^4*a2^6)+Sym(-40*a3^2*a5^12*a1^2) > +Sym(-36*a3^8*a2^4*a1^4)+Sym(-28*a1^12*a2^4) > +Sym(752*a4^8*a1^2*a3^2*a5^2*a2^2)+Sym(-70*a1^8*a4^8) > +Sym(272*a4^6*a3^4*a1^4*a5^2)+Sym(-344*a4^8*a3^4*a1^2*a5^2) > +Sym(8*a1^14*a4^2)+Sym(416*a3^2*a4^6*a1^6*a5^2) > +Sym(1520*a3^2*a4^4*a1^4*a5^4*a2^2)+Sym(-40*a1^8*a2^6*a5^2) > +Sym(-2008*a1^4*a2^4*a4^4*a3^4)+Sym(-a1^16)) * r^14 > +(Sym(1248*a1^4*a2^8*a4^4*a5^2)+Sym(-336*a1^2*a2^10*a4^2*a5^4) > +Sym(192*a2^8*a3^6*a4^4)+Sym(208*a4^6*a3^2*a2^8*a5^2) > +Sym(96*a1^6*a2^6*a3^6)+Sym(-160*a1^4*a2^6*a4^2*a5^2*a3^4) > +Sym(96*a3^4*a5^12*a1^2)+Sym(2544*a1^4*a2^8*a4^2*a5^2*a3^2) > +Sym(-2016*a2^2*a4^6*a3^4*a5^6)+Sym(320*a1^2*a3^8*a2^8) > +Sym(-288*a4^10*a2^4*a5^4)+Sym(-240*a4^10*a1^2*a2^6) > +Sym(2112*a4^6*a3^4*a2^4*a5^4)+Sym(-2880*a1^2*a4^2*a5^10*a2^2*a3^2) > +Sym(-8832*a1^4*a2^4*a4^2*a3^4*a5^4)+Sym(384*a1^2*a2^6*a3^6*a4^2*a5^2) > +Sym(144*a4^12*a3^2*a1^2*a5^2)+Sym(-16*a5^14*a3^2*a1^2)) * r^12 > +(Sym(-128*a1^6*a2^8*a4^4*a5^2)+Sym(-8320*a1^4*a2^8*a4^4*a5^2*a3^2) > +Sym(384*a2^10*a4^4*a1^2*a5^4)+Sym(-576*a3^8*a2^8*a4^4) > +Sym(5056*a1^6*a2^6*a4^2*a5^2*a3^4)+Sym(-3232*a1^6*a2^8*a4^2*a5^2*a3^2) > +Sym(5120*a1^2*a2^4*a3^4*a4^4*a5^6)+Sym(2976*a1^2*a2^10*a3^2*a4^2*a5^4) > +Sym(4928*a2^2*a4^6*a3^6*a5^6)+Sym(512*a4^6*a3^4*a2^4*a5^6) > +Sym(384*a4^10*a2^4*a5^6)+Sym(736*a1^10*a2^2*a4^6*a5^2) > +Sym(-5152*a1^4*a4^4*a3^4*a5^8)+Sym(-96*a3^4*a5^12*a1^4) > +Sym(-384*a4^8*a2^6*a5^6)+Sym(48768*a1^4*a2^4*a4^4*a3^4*a5^4) > +Sym(-1024*a2^2*a1^8*a4^8*a5^2)+Sym(224*a4^12*a3^2*a1^2*a5^2*a2^2) > +Sym(-256*a2^4*a3^2*a5^12*a1^2)+Sym(32*a2^2*a3^14*a1^2*a5^2)) * r^10 > +(Sym(4352*a1^6*a2^8*a4^4*a5^2*a3^2)+Sym(18176*a1^4*a2^8*a4^4*a5^2*a3^4) > +Sym(-768*a2^10*a4^6*a1^2*a5^4)+Sym(4096*a2^8*a1^8*a4^2*a5^2*a3^2) > +Sym(-38400*a1^4*a2^6*a4^4*a5^4*a3^4)+Sym(-1280*a4^6*a2^8*a1^2*a5^6) > +Sym(-1984*a1^2*a2^10*a3^2*a4^2*a5^6) > +Sym(-2304*a1^10*a2^2*a3^2*a4^4*a5^4) > +Sym(-11264*a1^6*a2^6*a4^4*a5^2*a3^4)+Sym(-256*a4^6*a3^10*a2^6) > +Sym(-16000*a1^2*a4^6*a3^6*a2^2*a5^6)+Sym(-3072*a4^6*a3^6*a2^4*a5^6) > +Sym(1024*a4^8*a2^4*a1^8*a5^2)+Sym(2048*a2^4*a1^8*a4^6*a5^4) > +Sym(-64*a4^14*a3^2*a1^2*a5^2*a2^2)+Sym(256*a2^4*a3^4*a5^12*a1^2) > +Sym(768*a2^4*a3^4*a5^10*a1^4)+Sym(512*a5^8*a3^8*a2^6)) * r^8 > +(Sym(-1024*a2^8*a1^8*a4^4*a5^4)+Sym(-8192*a1^6*a2^8*a4^4*a5^2*a3^4) > +Sym(-4096*a2^8*a1^8*a4^4*a5^2*a3^2)+Sym(-22272*a2^4*a1^8*a4^4*a5^4*a3^4) > +Sym(38912*a1^6*a2^6*a4^4*a5^4*a3^4)+Sym(512*a2^10*a3^2*a4^6*a5^6) > +Sym(2560*a4^6*a3^2*a2^8*a1^2*a5^6)+Sym(1536*a2^10*a4^4*a3^2*a1^2*a5^6) > +Sym(18432*a1^2*a4^6*a3^6*a2^4*a5^6)+Sym(2048*a1^6*a4^6*a2^6*a5^6) > +Sym(-256*a2^4*a3^4*a5^12*a1^4)+Sym(-256*a5^8*a3^8*a2^8)) * r^6 > +(Sym(8192*a4^6*a2^8*a1^4*a3^4*a5^4)+Sym(-1024*a2^10*a3^2*a4^6*a1^2*a5^6) > +Sym(4096*a2^8*a1^8*a4^4*a3^2*a5^4)+Sym(-12288*a1^6*a4^6*a2^6*a3^2*a5^6) > +Sym(-36864*a1^4*a2^6*a3^6*a4^4*a5^6)) * r^4 > +(Sym(-4096*a2^4*a1^8*a3^4*a4^8*a5^4) > +Sym(24576*a1^6*a2^6*a3^6*a4^4*a5^6)) * r^2 > + Sym(-16384*a1^6*a2^6*a3^6*a4^6*a5^6) = 0; > > where Sym stands short for symmetrization, > e.g., > Sym(-a1^16) = - a1^16 - a2^16 - a3^16 - a4^16 - a5^16, > Sym(-16384*a1^6*a2^6*a3^6*a4^6*a5^6) > = -16384*a1^6*a2^6*a3^6*a4^6*a5^6. > > Maybe someone will be able to spot a relation > between the exponents of the side lenghts ai > and the signs and coefficients in this formula, > which might be generalized to higher values of n ?! === Subject: Re: Program Synthesis and formallization? > Alan Smaill Charlie-Boo Alan Smaill: > why do you think a constructive proof is not a proof? > Because it doesn't demonstrate the truth of anything. >>So you claim; > do you believe in justifying your claims? > Then do so. > The claim is that what Curry-Howard describes is a proof. I say that >> in order to be a proof, then you have to demonstrate the truth of >> something. That has not been shown. >> CH makes a claim about a relationship between constructive proofs >> and computable functions. You object to the idea that the >> constructive proofs in question prove anything. >> Your claim, your justification. >> [You are trying to use the trick of requiring that someone prove a >> negative result, i.e., of the form there does not exist . . . >> That requires traversing the universal set, which may be impossible. >> We can only ask if one can give an example of what is supposed to >> exist, either a specific value or a set of values that must contain an >> example.] >> what statement there does not exist ... do you have in mind? > There does not exist an assertion that is being proven. You said that constructive proof doesn't demonstrate the truth of > anything. For any sentence p, there is a constructive proof of > p -> p. That is the proof of a true assertion, isn't it? So what is your claim about constructive logic? >> Do you claim constructive logic is unsound? > My claim is that there is no relationship between R(x,y) being a > recursive function R:x => y, and the assertion (all x)(exists y)R(x,y). This claim doesn't even mention constructive logic. > It doesn't help to see what you meant by not proving anything. As we have been though before, the claim is that there *is* > a relationship between constructive proof of (all x)(exists y)R(x,y) > and the existence of a recursive function f with the property that > (all x)R(x,f(x)). If you want to debunk CH, then *this* is the claim > that you need to object to. Your point is simply irrelevant > to the claim that CH depends on. > The problem with calling it a proof is that it implies that it can be > generated algorithmically no, it doesn't -- > of course it is undecidable in general whether a statement has a proof. > and that all the research into > theorem-proving applies and facilitates the implementation of the > system. This is not so. The lack of examples illustrates that. >> If so you just need some false conclusion derived from >> true premisses with constructive logic. >> Your claim, your justification. > .. > All right, here's an example: > find *two* proofs of >> [] ==> a:u(1)=>(a=>a=>a) >>in the system given earlier. >>You get back two *different* programs of this type, >>lambda(~,lambda(v0,lambda(~,v0))) > lambda(~,lambda(~, lambda(v0,v0))). >>According to CB, this proves that the system is not a compiler. > If you can show (1) the formal request (input) that produces the above >> programs, >> user input: >> [] ==> a:u(1)=>(a=>a=>a) >> (2) how these programs satisfy that formal request, >> In each case, evaluate (p of t1) where p is the program >> and t1 is a concrete type. The result is two different lambda terms >> that have the type (t1 => t1 => t1). The two functions have >> different input/output behaviour. >> (3) >> how the programs were formally generated from the formal request, >> automated proof search constructs one proof (with corresponding program); >> the user requests another proof, which has the corresponding different >> program. There are proof trees that can be inspected as in the >> earlier example for each of the proofs. > You are asserting that you can do something without giving an example > showing exactly what you do and how that produces the right answer. If you want the line by line proofs, say so; > it's excessive to keep posting these on a newsgroup, I think. Give the complete example, please. >> and >> (4) that the two programs use different algorithms (i.e. are not >> translatable into each other by a series of local changes), then you >> have shown that your system creates the algorithm, which is in fact >> more than what a compiler does. >> They have different I/O behaviour. >Have you ever read a description of how it works? What was the input > and output? The input is a series of program segments that are > combined into one program. * >>By CB so it must be true? > By CB, so no justification needed? > If you don't need to, then why should I have to? >> Oh, but I do. >> The last time someone showed Curry-Howard here, the user had to enter >> in pieces of the program, calling these pieces proofs. If you >> want me to dig it up . . . >> How does CBL synthesise the function that takes no input >> and return a value y such that 0 <= y*y < 1*1 ? > The user inputs: > (eA)MUL(x,x,A)^(eB)MUL(1,1,B)^~LT(A,0)^LT(A,B) > The inverses of the rules described in the ARXIV paper are applied to > reduce the problem down to axiomatized wffs e.g. MUL(I,J,x) [We can > multiply] which are then used to construct the program. I can give > more details if you wish. (The ARXIV paper goes into complete detail > for numerous examples, in Number Theeory, Database Retrieval, and > Theory of Computation.) Yes, I'd like to see the detail for this example. Will do, but not right now. (Uh-oh!) It creates a loop of values LT(x,1*1) from axiom LT(x,I) [We can list all numbers less than a given one.] using the QUIT rule, and checks each for being in 0 <= y*y by using axiom LT(I,J) [We can decide less than.] and rule NOT [The complement of a recursive set is recursive.] where we first determine y*y using axiom MUL(I,J,x) [We can multiply]. It also has to make the range of the loop include not y but y*y, so it uses DEF: MUL(a,b,c),MUL(a,b,c)^~LT(c,a). But I will have to sit down and work out the exact details to give the formal derivation. > Note that CBL doesn't produce any value that satisfies a given > predicate, but rather all values that satisfy that predicate. > a value y that meets a given predicate is not in general a function. > Your specific predicate is, but the general scheme of produce any > value that meets this predicate does not represent Program Synthesis > problems in general. > Program Synthesis deals with creating programs that produce a value > that is uniquesly defined: the value of a function, or all values in a > given set (the above.) This is an unnecessarily restricted view of program synthesis; > the specification may ask for a function where the output is constrained > in some way, but not uniquely defined. If CBL does not deal > with this, then it is not as general as CH. CH doesn't do . Sorry. You can't compare CBL to CH, only CBL to what CH claims. > CBL does that, as described in the ARXIV paper. > Does Curry-Howard? If the specification is chosen suitably, yes. C-B >> -- >> Alan Smaill > > -- > Alan Smaill === Subject: Re: Program Synthesis and formallization? > Because nobody has shown a Program Synthesis system before. Nobody can >> show that theirs works. OK how's this: you're right for an interpretation of 'Program > Synthesis' that pretty much nobody shares with you. Which is what? And the correct definition is what? My definition is that you input a statement that asserts what the program must do (e.g. in English, Set Theory, or Predicate Calculus - they're all the same) and it outputs one or more computer programs that do what you said. If you don't think that's Program Synthesis, then call it something else. (Long ago I developed this type of response to those who attempt to debate terminology. The point isn't that It does Program Synthesis, whatever is the correct definition of Program Synthesis., but rather, It creates computer programs, which people usually call Program Synthesis.) > Your definition may be the better one, Bad notion. Science must serve man, not vice-versa. > but you haven't really convinced me (or (I speak > presumptively) any of the other respondents here). ... >If we can make programs from >proofs, then we can make programs from the theorem to be proved, one >would think. > One would hope, one would like, one would have to consider that as an >extra step. Anyway there are many automated theorem provers that handle >the theorem to proof correspondence (or, as I expect you to counter, as >I claim wrongly). > That's the essence of the Curry-Howard bull. They say that they > can produce programs from proofs, and that proofs (naturally) can be > produced from theorems to be proven. But you have to show that all > works together. Neither of us is convincing the other of anything. Maybe another forum > would be more receptive to your ideas. Submit to CADE. That's for propositional calculus (Resolution) theorem-proving, I believe. TPTP is for general theorem-proving, but they give you the (BS - hardwired for each theorem) axioms to make it easier for their professor friends - which are also (wrongly) propositional calculus. My ARXIV paper gives 8 examples, describing each in detail, and I offer to answer any questions. If anyone did likewise for any other system, I would gladly be convinced. > Calling it a proof implies that it can be generated. That is part of > the Big Lie. > The only way to cut through this gibberish is to supply a complete > formal example. And by example, this means that it is a general scheme > that includes many program requirements, not just one quirky type > translation problem. Well, I think Alan Smaill went through a lot of trouble to show you one. No, it was the quirky type translation problem. After I asked about 5 times, he claimed that it could solve the real Program Synthesis problems of deciding and enumerating factors. And guess what happened when I asked for examples of programs that are generated to solve these problems? > Normally a prof giving a class would do an example (as part of his > duties as a prof) and expect the student to follow up with > reading/homework. Here (usenet) nobody owes anybody anything. Anyone who makes a contrary claim to anyone owes them a substantiation of that claim. (Read the U. S. Constitution.) > He's just > doing you (and anybody who is skeptical like you) a favor His unsubstantiated bull is a favor? How much do I owe for that favor? > because you > did not seem to be convinced by .the many examples that you can find on > just plop them in your lap). Example example? > You'll get many complete formal examples > (Curry-Howard, Program synthesis, theorem proving, etc, etc) supplying > them yourself by downloading and running the many apps out there. stupid enough to spend 5 days trying to download, install, learn, and use unsupported software to see what it does when he can read about it there's plenty of that on the internet. I agree! And plop is a good word for it. If Program Synthesis systems exist, then why do people waste time with Hurry-Coward which even its supporters say is only a first step and not an actual Program Synthesis system? Either you are saying that without any evidence or you are purposely holding back your examples. Which is it? > BTW Deciding what constitutes a program is extremely interesting. It > seems on the surface to be the same problem that Logicians have been > grappling with forever: Oh man, that is almost exactly what we've been trying to get at, that > there is not just an intuitive analogy between programs and proofs, but > a formal correspondence. No, I mean express vs. represent. It is easy to express a set but harder to write a program to generate it. It is the difference between the set of true sentences and the set of provable ones. It is trivial to express Fermat's Last Theorem, but took 350 years to represent it. And the correspondence is between wffs and programs, not proofs and programs. A wff represents a set (the set of values that when substituted for its free variable produces a provable sentence) and a program enumerates it. That is a fundamental result of the 1930's. Then Hurry-Coward came along and said that proofs, instead of wffs, are programs. How can people be so stupid as to not see the contradiction? > What is the difference between defining a set > by expressing it (the wff is true) vs. representing it (the wff is > provable)? how are expressing and representing different? (I think I understand > what you mean by true vs. provable, but I don't see how you want to > relate them to expressibility/representability. > The real answer should be a simple general principle. > Talking about how specific axioms, rules and wffs work misses the whole > point! I'm not sure what you're protesting about at here. Having a specific > set of axioms, etc at least gives a common point to help discuss the > general principles. The nature of truth vs. the nature of provability. Provability is easy. Truth is not. One thing I have noticed is that the paradoxes boil down to 3 mutually contradictory properties of formal systems. Proof systems have one (self-representability) while truth systems have another (complementability.) >You seem to consider the Curry-Howard isomorphism as trivial or some >other derogatory term but certainly not proper program synthesis. It is >definitely not all there is to program synthesis but is considered by >many to be an important part and to be non-trivial. That shows that actual Program Synthesis systems (aside from CBL) don't exist, otherwise why fool with Hurry-Coward? >guessed from responses to your posts, your attempt to convince others >will be an uphill battle against an army going down the other side. >That is mere demagoguery. >OK. ~OK. C-B > Mitch === Subject: Re: Program Synthesis and formallization? >Because nobody has shown a Program Synthesis system before. Nobody can >>show that theirs works. > OK how's this: you're right for an interpretation of 'Program > Synthesis' that pretty much nobody shares with you. Which is what? And the correct definition is what? My definition is that you input a statement that asserts what the > program must do (e.g. in English, Set Theory, or Predicate Calculus - > they're all the same) and it outputs one or more computer programs > that do what you said. If you don't think that's Program > Synthesis, then call it something else. (Long ago I developed this type of response to those who attempt to > debate terminology. The point isn't that It does Program > Synthesis, whatever is the correct definition of Program Synthesis., > but rather, It creates computer programs, which people usually call > Program Synthesis.) Great. That sounds like a good definition of Program Synthesis. And I'm with you on the point about terminology. Maybe the difference is in our usage. You said: nobody has shown a Program Synthesis system before and since I believe that they have, I infer that our ideas of what PS means must be different And because I believe that many other people agree with me, that's why I say that you have: ...an interpretation of 'Program Synthesis' that pretty much nobody shares with you. > Neither of us is convincing the other of anything. Maybe another forum > would be more receptive to your ideas. Submit to CADE. That's for propositional calculus (Resolution) theorem-proving, I > believe. No, it's a conference for all sorts of theorem proving: first order, 2nd, higher order, substructural logics, etc. Research in propositional systems has moved over (for political/administrative resource reasons) mostly to CAV or SAT > TPTP is for general theorem-proving, TPTP is a collection of theorems supported by a single research group, not a proving system (on the one hand), and not a vetting system (peer/expert review). I was implying that you should submit to CADE or some other related conference as the latter (usenet does not tend to give particularly robust reviews). TPTP is a place to get already-formed theorem statements to test on a newcomer's proving system (like for benchmarking). > but they give you the (BS - hardwired for each theorem) axioms to make it easier for their > professor friends so logical content is sometimes in axioms, sometimes in rules of inference; there's an efficiency tradeoff. as to hardwiring...that's a value judgement that, again, I think you share with hardly anybody. >- which are also (wrongly) propositional calculus. no, the TPTP axioms are first order. > My ARXIV paper gives 8 examples, describing each in detail, and I offer > to answer any questions. If anyone did likewise for any other system, > I would gladly be convinced. I think you're blinded by your interest in your own system. As advertisment, Otter/Mace is another excellent 1st order theorem prover. (I can't remember of there is a program synthesis part (like there is for nuPRL) but one can think of Mace (the finite model checker) as a way of calculating (finite) models for first order sentences). >Calling it a proof implies that it can be generated. That is part of >the Big Lie. > The only way to cut through this gibberish is to supply a complete >formal example. And by example, this means that it is a general scheme >that includes many program requirements, not just one quirky type >translation problem. > Well, I think Alan Smaill went through a lot of trouble to show you one. No, it was the quirky type translation problem. Another terminology/interpretation problem...one can always think of a correspondence (however complicated) as translation; you think it's trivial, others do not. > After I asked about 5 > times, he claimed that it could solve the real Program Synthesis > problems of deciding and enumerating factors. And guess what happened > when I asked for examples of programs that are generated to solve these > problems? I'd be curious to see such an example program, too, but it might take a lot of work on his part; he might be doing you a big favor by doing the work, or he may realize that his troubles aren't worth it (because every time he's come up with perfectly convincing examples, you don't sem to be convinced). Or there is certainly the possibility that absolutely no Program Synthesis system has ever been invented (other than yours of course) to take care of that problem. I personally don't know enough about all of them right now to support that latter statement. > Normally a prof giving a class would do an example (as part of his > duties as a prof) and expect the student to follow up with > reading/homework. Here (usenet) nobody owes anybody anything. Anyone who makes a contrary claim to anyone owes them a substantiation > of that claim. (Read the U. S. Constitution.) (I don't see how the Constitution is relevant). Right, burden of proof. You originally said: PS doesn't exist; others responded, yes, they do, look at X,Y,Z....; you said, that's not PS; others said, yes it is, here's an example; you said, no that's not an acceptable answer; others said, Hunh?, tried to explain why they are acceptable at length...; you said, no that's not acceptable... can we now ask you to support (take the burden of proof) why those reasons are not acceptable? > He's just > doing you (and anybody who is skeptical like you) a favor His unsubstantiated bull is a favor? How much do I owe for that > favor? Exactly. Nothing. You didn't like it? Well, you didn't pay anything for it anyway. > You'll get many complete formal examples > (Curry-Howard, Program synthesis, theorem proving, etc, etc) supplying > them yourself by downloading and running the many apps out there. stupid enough to spend 5 days trying to download, install, learn, and > use unsupported software to see what it does when he can read about it OK, so none of them are good enough for you. Fine. > If Program Synthesis systems exist, then why do people waste time with > Hurry-Coward which even its supporters say is only a first step > and not an actual Program Synthesis system? Either you are saying that without any evidence or you are purposely > holding back your examples. Which is it? Oh man, I'm really out of my mind. I forgot to mention LOPSTR (another conference), loko at papers there. >BTW Deciding what constitutes a program is extremely interesting. It >seems on the surface to be the same problem that Logicians have been >grappling with forever: > Oh man, that is almost exactly what we've been trying to get at, that > there is not just an intuitive analogy between programs and proofs, but > a formal correspondence. No, I mean express vs. represent. It is easy to express a set but > harder to write a program to generate it. It is the difference between > the set of true sentences and the set of provable ones. It is trivial > to express Fermat's Last Theorem, but took 350 years to represent it. hmm... I didn't understand how you meant 'represent'. It sounds like 'proof'. > And the correspondence is between wffs and programs, not proofs and > programs. Which correspondence? CH talks about a correspondence between proofs and programs, plain and simple. Maybe you're saying that the CH correspondence is not the important one people should be looking at? And that your (new?) wff-program is more important? > A wff represents a set (the set of values that when > substituted for its free variable produces a provable sentence) and a > program enumerates it. Are you defining wff anew here (i.e. let's define a wff to be a set with these properties)...? or are you just saying one can represent a set using a wff? (I tend to think of a 'wff' as simply a grammar) Mitch === Subject: Re: Program Synthesis and formallization? >Because nobody has shown a Program Synthesis system before. Nobody can >>show that theirs works. > OK how's this: you're right for an interpretation of 'Program > Synthesis' that pretty much nobody shares with you. Which is what? And the correct definition is what? My definition is that you input a statement that asserts what the > program must do (e.g. in English, Set Theory, or Predicate Calculus - > they're all the same) and it outputs one or more computer programs > that do what you said. If you don't think that's Program > Synthesis, then call it something else. (Long ago I developed this type of response to those who attempt to > debate terminology. The point isn't that It does Program > Synthesis, whatever is the correct definition of Program Synthesis., > but rather, It creates computer programs, which people usually call > Program Synthesis.) Great. That sounds like a good definition of Program Synthesis. And I'm with you on the point about terminology. Maybe the difference is in our usage. You said: nobody has shown a Program Synthesis system before and since I believe that they have, I infer that our ideas of what PS means must be different And because I believe that many other people agree with me, that's why I say that you have: ...an interpretation of 'Program Synthesis' that pretty much nobody shares with you. > Neither of us is convincing the other of anything. Maybe another forum > would be more receptive to your ideas. Submit to CADE. That's for propositional calculus (Resolution) theorem-proving, I > believe. No, it's a conference for all sorts of theorem proving: first order, 2nd, higher order, substructural logics, etc. Research in propositional systems has moved over (for political/administrative resource reasons) mostly to CAV or SAT > TPTP is for general theorem-proving, TPTP is a collection of theorems supported by a single research group, not a proving system (on the one hand), and not a vetting system (peer/expert review). I was implying that you should submit to CADE or some other related conference as the latter (usenet does not tend to give particularly robust reviews). TPTP is a place to get already-formed theorem statements to test on a newcomer's proving system (like for benchmarking). > but they give you the (BS - hardwired for each theorem) axioms to make it easier for their > professor friends so logical content is sometimes in axioms, sometimes in rules of inference; there's an efficiency tradeoff. as to hardwiring...that's a value judgement that, again, I think you share with hardly anybody. >- which are also (wrongly) propositional calculus. no, the TPTP axioms are first order. > My ARXIV paper gives 8 examples, describing each in detail, and I offer > to answer any questions. If anyone did likewise for any other system, > I would gladly be convinced. I think you're blinded by your interest in your own system. As advertisment, Otter/Mace is another excellent 1st order theorem prover. (I can't remember of there is a program synthesis part (like there is for nuPRL) but one can think of Mace (the finite model checker) as a way of calculating (finite) models for first order sentences). >Calling it a proof implies that it can be generated. That is part of >the Big Lie. > The only way to cut through this gibberish is to supply a complete >formal example. And by example, this means that it is a general scheme >that includes many program requirements, not just one quirky type >translation problem. > Well, I think Alan Smaill went through a lot of trouble to show you one. No, it was the quirky type translation problem. Another terminology/interpretation problem...one can always think of a correspondence (however complicated) as translation; you think it's trivial, others do not. > After I asked about 5 > times, he claimed that it could solve the real Program Synthesis > problems of deciding and enumerating factors. And guess what happened > when I asked for examples of programs that are generated to solve these > problems? I'd be curious to see such an example program, too, but it might take a lot of work on his part; he might be doing you a big favor by doing the work, or he may realize that his troubles aren't worth it (because every time he's come up with perfectly convincing examples, you don't sem to be convinced). Or there is certainly the possibility that absolutely no Program Synthesis system has ever been invented (other than yours of course) to take care of that problem. I personally don't know enough about all of them right now to support that latter statement. > Normally a prof giving a class would do an example (as part of his > duties as a prof) and expect the student to follow up with > reading/homework. Here (usenet) nobody owes anybody anything. Anyone who makes a contrary claim to anyone owes them a substantiation > of that claim. (Read the U. S. Constitution.) (I don't see how the Constitution is relevant). Right, burden of proof. You originally said: PS doesn't exist; others responded, yes, they do, look at X,Y,Z....; you said, that's not PS; others said, yes it is, here's an example; you said, no that's not an acceptable answer; others said, Hunh?, tried to explain why they are acceptable at length...; you said, no that's not acceptable... can we now ask you to support (take the burden of proof) why those reasons are not acceptable? > He's just > doing you (and anybody who is skeptical like you) a favor His unsubstantiated bull is a favor? How much do I owe for that > favor? Exactly. Nothing. You didn't like it? Well, you didn't pay anything for it anyway. > You'll get many complete formal examples > (Curry-Howard, Program synthesis, theorem proving, etc, etc) supplying > them yourself by downloading and running the many apps out there. stupid enough to spend 5 days trying to download, install, learn, and > use unsupported software to see what it does when he can read about it OK, so none of them are good enough for you. Fine. > If Program Synthesis systems exist, then why do people waste time with > Hurry-Coward which even its supporters say is only a first step > and not an actual Program Synthesis system? Either you are saying that without any evidence or you are purposely > holding back your examples. Which is it? Oh man, I'm really out of my mind. I forgot to mention LOPSTR (another conference), loko at papers there. >BTW Deciding what constitutes a program is extremely interesting. It >seems on the surface to be the same problem that Logicians have been >grappling with forever: > Oh man, that is almost exactly what we've been trying to get at, that > there is not just an intuitive analogy between programs and proofs, but > a formal correspondence. No, I mean express vs. represent. It is easy to express a set but > harder to write a program to generate it. It is the difference between > the set of true sentences and the set of provable ones. It is trivial > to express Fermat's Last Theorem, but took 350 years to represent it. hmm... I didn't understand how you meant 'represent'. It sounds like 'proof'. > And the correspondence is between wffs and programs, not proofs and > programs. Which correspondence? CH talks about a correspondence between proofs and programs, plain and simple. Maybe you're saying that the CH correspondence is not the important one people should be looking at? And that your (new?) wff-program is more important? > A wff represents a set (the set of values that when > substituted for its free variable produces a provable sentence) and a > program enumerates it. Are you defining wff anew here (i.e. let's define a wff to be a set with these properties)...? or are you just saying one can represent a set using a wff? (I tend to think of a 'wff' as simply a grammar) Mitch === Subject: Re: Program Synthesis and formallization? anything. For any sentence p, there is a constructive proof of > p -> p. That is the proof of a true assertion, isn't it? A set of axioms and rules that creates all possible sentences (e.g. just follow the definition of the syntax of a wff in general) also includes true assertions such as p->p, but is that proving anything? > So what is your claim about constructive logic? > My claim is that there is no relationship between R(x,y) being a > recursive function R:x => y, and the assertion (all x)(exists y)R(x,y). This claim doesn't even mention constructive logic. That's how Curry-Howard works, and they call it constructive logic. > As we have been though before, the claim is that there *is* > a relationship between constructive proof of (all x)(exists y)R(x,y) > and the existence of a recursive function f with the property that > (all x)R(x,f(x)). If you want to debunk CH, then *this* is the claim > that you need to object to. Your point is simply irrelevant > to the claim that CH depends on. That's what I just said, but that there is no such relationship. What about R(x,y) <=> x=(y*y)? Does Curry-Howard generate a program to compute R: x=>y? > The problem with calling it a proof is that it implies that it can be > generated algorithmically no, it doesn't -- A fundamental result of Mathematical Logic is that the theorems of an axiomatic system form a recursively enumerable set. (In fact, the class of representable sets is exactly the recursively enumerable ones.) > of course it is undecidable in general whether a statement has a proof. To be generated algorithmically means to be recursively enumerable, not recursive (decidable.) > and that all the research into > theorem-proving applies and facilitates the implementation of the > system. This is not so. The lack of examples illustrates that. >> If so you just need some false conclusion derived from >> true premisses with constructive logic. >> Your claim, your justification. See The CB Book of Fundamental Results of Logic for Newbies. I think it might clear up some of your above confusion. (Note: I am NOT referencing it just to increase my royalties! There are other reasons as well.) C-B > -- > Alan Smaill === Subject: Re: Program Synthesis and formallization? > You said that constructive proof doesn't demonstrate the truth of >> anything. For any sentence p, there is a constructive proof of >> p -> p. That is the proof of a true assertion, isn't it? A set of axioms and rules that creates all possible sentences (e.g. > just follow the definition of the syntax of a wff in general) also > includes true assertions such as p->p, but is that proving anything? It is a proof system, indeed; but it's not sound. Are you claiming that constructive logic is unsound? >> So what is your claim about constructive logic? > My claim is that there is no relationship between R(x,y) being a >> recursive function R:x => y, and the assertion (all x)(exists y)R(x,y). >> This claim doesn't even mention constructive logic. That's how Curry-Howard works, and they call it constructive > logic. So please rephrase your claim to make explicit the connection with constructive proof. >> As we have been though before, the claim is that there *is* >> a relationship between constructive proof of (all x)(exists y)R(x,y) >> and the existence of a recursive function f with the property that >> (all x)R(x,f(x)). If you want to debunk CH, then *this* is the claim >> that you need to object to. Your point is simply irrelevant >> to the claim that CH depends on. > That's what I just said, but that there is no such relationship. And your claim is simply irrelevant. > What about R(x,y) <=> x=(y*y)? Does Curry-Howard generate a program to > compute R: x=>y? Standard CH deals with provably total recursive functions. With systems like Nuprl, partial functions can also be synthesised. But you are simply ignoring my claim that when (all x)(exists y)R(x,y) is constructively provable, then there is a recursive function f with the property that (all x)R(x,f(x)). Your total silence on this point is noted. >> The problem with calling it a proof is that it implies that it can be >> generated algorithmically >> no, it doesn't -- The idea that proofs found by hand are not proofs at all is just weird. > A fundamental result of Mathematical Logic is that the theorems of an > axiomatic system form a recursively enumerable set. (In fact, the > class of representable sets is exactly the recursively enumerable > ones.) > of course it is undecidable in general whether a statement has a proof. To be generated algorithmically means to be recursively > enumerable, not recursive (decidable.) In that case, of course the proofs in CH are generated algorithmically. Why ever would you doubt that? >> and that all the research into >> theorem-proving applies and facilitates the implementation of the >> system. This is not so. The lack of examples illustrates that. > If so you just need some false conclusion derived from > true premisses with constructive logic. >> Your claim, your justification. See The CB Book of Fundamental Results of Logic for Newbies. I > think it might clear up some of your above confusion. Still your claim, your justification. > (Note: I am NOT referencing it just to increase my royalties! There > are other reasons as well.) C-B > -- >> Alan Smaill > -- Alan Smaill === Subject: Re: Program Synthesis and formallization? > You said that constructive proof doesn't demonstrate the truth of >> anything. For any sentence p, there is a constructive proof of >> p -> p. That is the proof of a true assertion, isn't it? > A set of axioms and rules that creates all possible sentences (e.g. > just follow the definition of the syntax of a wff in general) also > includes true assertions such as p->p, but is that proving anything? It is a proof system, indeed; > but it's not sound. Then they're all proof systems and the question of proving anything is meaningless. > Are you claiming that constructive logic is unsound? >> So what is your claim about constructive logic? >> My claim is that there is no relationship between R(x,y) being a >> recursive function R:x => y, and the assertion (all x)(exists y)R(x,y). >> This claim doesn't even mention constructive logic. > That's how Curry-Howard works, and they call it constructive > logic. So please rephrase your claim to make explicit the connection with > constructive proof. You already know what it is. >> As we have been though before, the claim is that there *is* >> a relationship between constructive proof of (all x)(exists y)R(x,y) >> and the existence of a recursive function f with the property that >> (all x)R(x,f(x)). If you want to debunk CH, then *this* is the claim >> that you need to object to. Your point is simply irrelevant >> to the claim that CH depends on. > That's what I just said, but that there is no such relationship. And your claim is simply irrelevant. > What about R(x,y) <=> x=(y*y)? Does Curry-Howard generate a program to > compute R: x=>y? Standard CH deals with provably total recursive functions. > With systems like Nuprl, partial functions can also be synthesised. And what does it have to prove (or otherwise)? > But you are simply ignoring my claim that when (all x)(exists y)R(x,y) > is constructively provable, then there is a recursive function f with the > property that (all x)R(x,f(x)). Your total silence on this point is noted. At least someone is paying attention. Answer the above question to get closer to the truth. >> The problem with calling it a proof is that it implies that it can be >> generated algorithmically >> no, it doesn't -- The idea that proofs found by hand are not proofs at all > is just weird. Ok, it's by hand. Then it's not Program Synthesis. > A fundamental result of Mathematical Logic is that the theorems of an > axiomatic system form a recursively enumerable set. (In fact, the > class of representable sets is exactly the recursively enumerable > ones.) >> of course it is undecidable in general whether a statement has a proof. > To be generated algorithmically means to be recursively > enumerable, not recursive (decidable.) In that case, of course the proofs in CH are generated algorithmically. > Why ever would you doubt that? Just give an example of a Program Synthesis problem solved by C-H. No extra input. No phony output a program that outputs a program examples. Real life Program Synthesis. Real Program Synthesis systems write programs that do mathematics! >> and that all the research into >> theorem-proving applies and facilitates the implementation of the >> system. This is not so. The lack of examples illustrates that. >> If so you just need some false conclusion derived from > true premisses with constructive logic. >>Your claim, your justification. > See The CB Book of Fundamental Results of Logic for Newbies. I > think it might clear up some of your above confusion. Still your claim, your justification. Oh, you want a free copy? Autographed? (If I gave all of them away free, how would I pay my rent?) C-B > (Note: I am NOT referencing it just to increase my royalties! There > are other reasons as well.) > -- > Alan Smaill === Subject: Re: Program Synthesis and formallization? Calling it a proof implies that it can be generated. That is part of > the Big Lie. Show me that program of yours that generates CBL > proofs. The above post, from 3/26/03, lists 505 incompleteness theorems, which are siblings to, and include, theorems of Godel and Rosser, and explains how they were generated. You are free to speculate as to how you believe they were actually created. What is the value of programming a system, such as Curry-Howard, if it has not been shown to produce anything? Curry-Howard is often called a first step toward Program Synthesis. But that is only speculation and history belies that. After decades, it is still at the first step. Any fair-mined, honest person would admit that the facts show us that it has failed. C-B > Oh, no, wait! We've been there before, haven't we, > Charlie? You can't show this program, because it's > not written in PHP. LOL. So, if this proofs can be generated thing doesn't > even apply to your own little petty system, why is > it important that it applies to Curry-Howard based > systems? (Btw. I assume that by generate you mean > something more intelligent than just simply > enumerating proofs.) >groente > -- Sander === Subject: Re: Program Synthesis and formallization? >> Show me that program of yours that generates CBL >> proofs. The above post, from 3/26/03, lists 505 incompleteness theorems, which > are siblings to, and include, theorems of Godel and Rosser, and > explains how they were generated. But what it doesn't contain are the generated proofs of these theorems. And also this proof generator of yours is missing. In fact, it doesn't even contain an informal description of the algorithm used to generate the proofs. So, no, I'm not impressed. > You are free to speculate as to how > you believe they were actually created. Ah... I may speculate. Does that mean that you have no intention to actually substantiate your own claims? The only thing you need to do is post a link to the source code of this proof generator of yours. >What is the value of programming a system, such as Curry-Howard, if it > has not been shown to produce anything? What is the value of a programming system, such as CB's proof generator, if it has not been shown to actually *exist*? groente -- Sander === Subject: Re: Program Synthesis and formallization? > Show me that program of yours that generates CBL >> proofs. > > The above post, from 3/26/03, lists 505 incompleteness theorems, which > are siblings to, and include, theorems of Godel and Rosser, and > explains how they were generated. But what it doesn't contain are the generated proofs of these > theorems. And also this proof generator of yours is missing. > In fact, it doesn't even contain an informal description of > the algorithm used to generate the proofs. Click on the link. > So, no, I'm not impressed. > You are free to speculate as to how > you believe they were actually created. Ah... I may speculate. Does that mean that you have no > intention to actually substantiate your own claims? The > only thing you need to do is post a link to the source > code of this proof generator of yours. complete program listing of a simplified version. It is available if you search for it. I pointed this out before, but the person to whom I said this replied that he refuses to pay the cost of the reprint. Whose ass do I need to lick? > What is the value of programming a system, such as Curry-Howard, if it > has not been shown to produce anything? What is the value of a programming system, such as CB's > proof generator, if it has not been shown to actually > *exist*? Then I am quite the magician! C-B >groente > -- Sander === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If you cannot construct it, if the procedure is non-effective, how > can you claim you have created a new number, to be added to the list? It bugs me... Any problem there is in the metaphorical descriptions you use, such as 'created', 'here's what you do', 'do it', etc. There is no doing or creating in the proof. The proof is of the existence of a certain function. We don't have to say we 'created' the function or that there was any 'doing' involved. We just need to prove the function exists. MoeBlee === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If you cannot construct it, if the procedure is non-effective, how > can you claim you have created a new number, to be added to the list? It isn't a procedure and you don't have to create this new number. You merely have to convince yourself it exists. The first important point is that any sequence of digits after a decimal point, 0.xxxxx... represents a valid value in [0,1]. The remainder of the proof is to construct an argument that there exists at least one such sequence that isn't on the list. You don't have to get to the 10^1000^1000-th digit for instance. Does the list have a 10^1000^1000-th number? Of course. Does that number have a 10^1000^1000-th digit? Of course. Then there at least 8 possible values that our hypothetical number might have in that position. (One variant of the procedure would be to avoid 9's, so that we aren't ever considering numbers that end in ...999...). So what we are doing is establishing a set of properties which this hypothetical number must have. If it has those properties, it isn't on the original list. Does any number have those properties? Yes, uncountably many numbers do (8 choices for 1st digit, 8 choices for 2nd digit..). Once you've convinced yourself that the set of such numbers is non-empty, the proof is done. You don't have to actually generate one number. === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the > diagonalization > bit, a consideration I have never seen addressed. > You hypothesize the countable list of all reals, > encoded > in binary, then diagonalize out of it, by > complementing the > diagonal. Sounds good. >... >No, binary doesn't work. >Ross > Why not? === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If you cannot construct it, if the procedure is non-effective, how > can you claim you have created a new number, to be added to the list? It bugs me... Mark There is a version of the diagonal argument that deals with effective procedures. It proves that the set of computable real numbers is not effectively countable. Given an effective enumeration of a set of computable real numbers, you can construct a computable real number not in the list, using the diagonal method. === Subject: Re: Cantor's diagonalization argument In sci.logic, Mark-T on 26 Oct 2006 20:53:55 -0700 > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If you cannot construct it, if the procedure is non-effective, how > can you claim you have created a new number, to be added to the list? It bugs me... Mark > First, the new number is not added to the list; it's provably not *in* the list to begin with and therefore proves that not all reals can be thus ordered [*]. Second, the computability of the diagonal number, given the computability of the original numbers, is not really an issue; all one needs is the list of decimal expansions. If one uses Cauchuy, one can always take the n'th digit of a number by extending the series far enough. Dedekind cuts are a little tougher and I'd have to research the issue. And finally, there's an alternate proof that the reals are uncountable as well -- Cantor's *first* proof. As for the encoding -- binary doesn't quite work, but one can work around that issue by using base 4 instead. It's the old 0.111... = 1.000... thang. [*] If one has a list L of reals, and a real r not in the list, one can add r to the list, but that just generates another list L', which can then be used to generate another diagonal number r' -- this can continue indefinitely. -- #191, ewill3@earthlink.net Windows Vista. It'll Fix Everything(tm). -- === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. >You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. >Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If it bothers you that you can't create this number in finite time, does it bother you that you can create the integers in finite time, either? If you reject a mathematical object because you can't create it in finite time, it seems to me you are rejecting the infinite. If you reject the infinite, you have no need of any theory of cardinality of infinite sets, Cantor's or otherwise. But, please, let the rest of us get on with the infinite sets we have come to know and use. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Cantor's diagonalization argument > Something has always bothered me about the diagonalization > bit, a consideration I have never seen addressed. You hypothesize the countable list of all reals, encoded > in binary, then diagonalize out of it, by complementing the > diagonal. Sounds good. Except... isn't there a slight problem here, in the sense of > 'computable', or 'effective procedure'? How does one diagonalize > an infinite set? Of course, it's easy to describe, here's what > you do, but in fact you cannot DO it! You cannot create this > new 'unlisted' number, not in finite time, anyhow... If you cannot construct it, if the procedure is non-effective, how > can you claim you have created a new number, to be added to the list? It bugs me... Mark > By this argument, sqrt(2) isn't a Real, because there is no effective procedure to generate its decimal expansion in finite time. However, there is a process to generate the nth decimal place of sqrt(2) for all n - just as there is for the diagonal number. Its no different in this respect to any other number with an infinite decimal expansion (including 1/3). === Subject: Re: Cantor's diagonalization argument > By this argument, sqrt(2) isn't a Real, because there is no effective > procedure to generate its decimal expansion in finite time. >However, there is a process to generate the nth decimal place of sqrt(2) for > all n - just as there is for the diagonal number. Its no different in this > respect to any other number with an infinite decimal expansion (including > 1/3). Whether or not there is an effective procedure for generating the nth decimal of the diagonal number depends entirely on the list of which it is the diagonal of. For example, there is no effective procedure for generating the nth decimal of the diagonal of the list of reals given by R_i = sum of 10^-n*a_n,i over all n in N where a_n,i is 0 iff the nth sentence in the language of set theory (in some fixed enumeration) is true when interpreted in V_omega+i, and 0 if not. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument >> If you cannot construct it, if the procedure is non-effective, how >> can you claim you have created a new number, to be added to the list? >You have a rule which even a Turing machine can follow to any desired > degree of precision. What rule? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument >> If you cannot construct it, if the procedure is non-effective, how >> can you claim you have created a new number, to be added to the list? >You have a rule which even a Turing machine can follow to any desired > degree of precision. >What rule? Any of the various rules for constructing the Cantor diagonal given a list of reals in which the nth decimal digit of the nth listed number can be determined in a finite number of steps. === Subject: Re: Cantor's diagonalization argument > >> What rule? >Any of the various rules for constructing the Cantor diagonal given a > list of reals in which the nth decimal digit of the nth listed number > can be determined in a finite number of steps. In that case, sure. But most lists of reals are such that the nth decimal digit of the nth listed number can not be determined in a finite number of steps. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument > >> What rule? >Any of the various rules for constructing the Cantor diagonal given a > list of reals in which the nth decimal digit of the nth listed number > can be determined in a finite number of steps. >In that case, sure. But most lists of reals are such that the nth > decimal digit of the nth listed number can not be determined in a finite > number of steps. Most in what sense? There are certainly uncountably many reals which , for any fixed natural n, can have their nth decimal digit determined in a finite number of steps, those between 0 and 1/10^(n+1), for example. === Subject: Re: Cantor's diagonalization argument > If you cannot construct it, if the procedure is non-effective, how >> can you claim you have created a new number, to be added to the list? > You have a rule which even a Turing machine can follow to any desired > degree of precision. What rule? A better question is what machine? -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument > You have a rule which even a Turing machine can follow to any desired > degree of precision. >> What rule? > A better question is what machine? Why is that a better question? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument >> You have a rule which even a Turing machine can follow to any desired >> degree of precision. > What rule? >> >> A better question is what machine? Why is that a better question? Why is Eliza not a better machine? Lee Rudolph === Subject: Re: Cantor's diagonalization argument You have a rule which even a Turing machine can follow to any desired > degree of precision. >> What rule? > A better question is what machine? Why is that a better question? Because the QM cranks use virtual machines, -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument >> Why is that a better question? > Because the QM cranks use virtual machines, If you say so. I rather doubt Virgil is a QM crank, though. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument They are joshing you. Virgil's rule was the rule used in diagonalizing: if the nth digit of the nth number is, say, 5, use 7, if not use 5. ZZBunker's statement that the question What is the Turing machine is more important was due to the fact that the rule is necessarily included in the machine and the whole question was whether such a machine exists. Since it does, the process is calculable. === Subject: Re: Cantor's diagonalization argument >> Why is that a better question? > Because the QM cranks use virtual machines, >If you say so. I rather doubt Virgil is a QM crank, though. === Subject: Re: Simple Set Theory question If you are willing to work simply from an injection (and get >equinumerousity by Shroeder-Bernstein), use a Goedel >numbering: > {a,b,c,...z} |--> 2^a * 3^b * ... p^z > Right, of course that is the usual method. But I said that I want to do > this without having the prime factorization at my disposal. Doesn't the > method you mention presume that we've already proven the prime > factorization theorem? It relies on the uniqueness of prime factorization (but not necessarily > on existence) and on the infinitude of primes to prove injectivity. Of > course any appeal to Shroeder-Bernstein would pretty well eclipse > quibbling over the proof of the Fundamental Thm. of Arithmetic: Not really. I have Schroder-Bernstein in place before fundamental theorem of arithmetic. > I suppose binary expansions are a simpler and more natural approach, > yielding a bijection between positive integers and nonempty finite > subsets of the positive integers: Okay, I'm getting the feeling that I might have to reconsider my goal of trying this without fundamental theorem of arithmetic or basis reperesenatation already proven. MoeBlee === Subject: Re: Simple Set Theory question to the location of the ones. Can you not prove that every natural > number has a unique expression in base 2? Oops. I spaced and didn't even recognize the basis representation theorem in action. But I'm trying to prove the finite subsets theorem without having basis representation and prime factorization already proven. Sorry for driving such a hard bargain; it just has to do with the way I'm ordering the trail of theorems. MoeBlee === Subject: Re: Infinity equals one > Hey, does anybody know how to prove that infinity > equals one, or how I > would get started on the proof. For example, consider E={ {1,oo},{a}} (a<>1,a<>oo). Then, A={1,oo} B={a} is a partition of E and therefore defines an equivalence relation so 1 and oo define the same equivalence class and in some sense 1=oo. This is the best I can do for you. There are other possibilities but all of them including the foregoing are useless. Fernando. === Subject: Re: Infinity equals one > Hey, does anybody know how to prove that infinity > equals one, or how I > would get started on the proof. > For example, consider E={ {1,oo},{a}} (a<>1,a<>oo). >Then, A={1,oo} B={a} is a partition of E and > therefore > defines an equivalence relation so 1 and oo define > the > same equivalence class and in some sense 1=oo. >This is the best I can do for you. There are other > possibilities but all of them including the foregoing > are > useless. >Fernando. Of course I meant E={1,oo,a}. Fernando === Subject: Re: Infinity equals one > Hey, does anybody know how to prove that infinity equals one, or how I > would get started on the proof. State it as an axiom. :) TOny === Subject: Re: Infinity equals one > Hey, does anybody know how to prove that infinity equals one, or how I > would get started on the proof. > State it as an axiom. :) >That was already suggested. :>/ >MoeBlee > Yeah, I saw your response right after I posted that. I was thinking of you... :) TOEknee === Subject: Re: Infinity equals one <4540e765$1@news2.lightlink> <45422df4$1@news2.lightlink Yeah, I saw your response right after I posted that. I was thinking of > you... :) Yeah, you know it just had to be one of those religionistic followers of status quo Cantorian transfinitology. MoeBlee === Subject: Re: An uncountable countable set <9pkkj2dgbdk0ok66d3v9b6l3evdoa51id1@4ax> By having read a proof. A proof that there can be no consistent theory . . .. Truly > fascinating. Do tell us more about this proof. I said no such thing as that there is a proof that there is no consistent theory. I say something, and you come back to tell me I said something completely different. You are the winner! > Arithmetic: 1, 2, 3 . . . The calculus: disintegration and integration > of definite integrals. Infinity as the number of infinitesimals. Do I attempt a telepathic tap into your brain for all of that mathematics, or do you recommend a book I can read that has all that mathematics (especially 'infinity as the number of infinitesimals') but avoids all the nasty neo-mathematiker propaganda? Or should I just think it for myself and carry it around privately in my brain the way you do? MoeBlee === Subject: Re: An uncountable countable set <452d8ef0@news2.lightlink> <452e55b6@news2.lightlink> <452ef2bc@news2.lightlink> <452fc2e7@news2.lightlink> <45310346@news2.lightlink> <45319b8c@news2.lightlink> <453434b7@news2.lightlink> <4535970a@news2.lightlink> <453d9bb4@news2.lightlink> You have agreed with everything so far. At every point before noon balls >> remain. You claim nothing changes at noon. Is there something between >> noon and before noon, when those balls disappeared? If not, then they >> must still be in there. > Of course there is a something between before noon and noon where >> each ball disappears. At step n, time 2^-n min before noon, ball n is >> removed. This happens for every ball, since there is a step n for >> every ball. The balls are removed, one by one, one at each step, >> before noon. As each ball n is removed, how many remain? Can any be removed and leave > an empty vase? Each ball n is placed into the vase at time 2^int(n/10), and then later removed at time n. This happens for every ball before noon. So every ball is inserted and then later removed from the vase before noon. At any given time n before noon, ten balls are added to the vase and then ball n (which was added to the vase in a previous step) is removed. Your entire confusion results from assuming a last time prior to noon, but there is no such time. === Subject: Re: An uncountable countable set > You have agreed with everything so far. At every point before noon balls > remain. You claim nothing changes at noon. Is there something between > noon and before noon, when those balls disappeared? If not, then they > must still be in there. >Of course there is a something between before noon and noon where > each ball disappears. At step n, time 2^-n min before noon, ball n is > removed. This happens for every ball, since there is a step n for > every ball. The balls are removed, one by one, one at each step, > before noon. > >> As each ball n is removed, how many remain? Can any be removed and leave >> an empty vase? >Each ball n is placed into the vase at time 2^int(n/10), and then later > removed at time n. This happens for every ball before noon. So every > ball is inserted and then later removed from the vase before noon. >At any given time n before noon, ten balls are added to the vase and > then ball n (which was added to the vase in a previous step) is > removed. Your entire confusion results from assuming a last time > prior to noon, but there is no such time. > At no time prior to noon are all balls removed. Nor are any removed at noon. It cannot be empty, then. === Subject: Re: An uncountable countable set >>For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >>TOE-Knee >> What is this IFR, inverse function rule? I've heard you mention it. That which can be done; can be undone. Not in the case of sex. ~v~~ === Subject: Re: An uncountable countable set >For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >> TOE-Knee >> What is this IFR, inverse function rule? I've heard you mention it. >> That which can be done; can be undone. >Not in the case of sex. >~v~~ Oh God, tell me about it. ;) 01oo ~v~~ === Subject: Re: An uncountable countable set >> even more sheer brilliance! <... >> I won't quote more that opening and close, as one can become >> overwhelmed by so much wisdom from just one man in just one day. Ah, I see you've noticed. Meanwhile, Have you tried searching the >archive for Zick + transcendental? > Ah, Brian, ever the amanuensis. ~v~~ === Subject: Re: An uncountable countable set > Ah, Brian, ever the amanuensis. Zick, ever the nuisance. Ah, Moe, truth is often a nuisance. ~v~~ === Subject: Re: An uncountable countable set >Ah, Brian, ever the amanuensis. >> Zick, ever the nuisance. >Ah, Moe, truth is often a nuisance. >~v~~ Stop confusing me with facts, Lester. Not and/or the inverse of not the notness of it all. I just don't get it. :) TOny === Subject: Re: An uncountable countable set > Stop confusing me with facts, Lester. TO is easily confused by facts. But Lester rarely provides any. A marriage made in Heaven? Or Hell? === Subject: Re: An uncountable countable set > Ah, Moe, truth is often a nuisance. >Then you're as much of a nuisance as is a cool breeze on a sunny spring > day, as a cleansing and quenching rain that ends a drought, as a > magnificent symphony orchestra heard in an amphitheatre of impeccable > acoustics. >Moe Blee > As a kitty cat on your lap, on a chilly night... :) Lesterrrrrrrr...... Actually, I named my last cat Lester, before I met Lester here online. I had to scrape his pieces off the road.... === Subject: Re: An uncountable countable set <4542ac76@news2.lightlink >> Ah, Moe, truth is often a nuisance. > Then you're as much of a nuisance as is a cool breeze on a sunny spring > day, as a cleansing and quenching rain that ends a drought, as a > magnificent symphony orchestra heard in an amphitheatre of impeccable > acoustics. > Moe Blee >As a kitty cat on your lap, on a chilly night... :) Ah, sweet. > Actually, I named my last cat Lester, before I met Lester here online. I > had to scrape his pieces off the road.... Ouch, not so sweet. MoeBlee === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Surely fodder for a Jesse F. Hughes tagline. MoeBlee === Subject: Re: An uncountable countable set >> I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. >Surely fodder for a Jesse F. Hughes tagline. >MoeBlee > Oh, surely. === Subject: Re: An uncountable countable set > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) >> TOE-Knee >> What is this IFR, inverse function rule? I've heard you mention it. >That which can be done; can be undone. > The inverse mapping gives the number of elements over a given value range, which can be applied to the entire set of reals, providing intuitive results (like half as many events as naturals) for infinite sets when that real range is considered constant. Tony === Subject: Re: An uncountable countable set > The inverse mapping gives the number of elements over a given value > range, which can be applied to the entire set of reals, providing > intuitive results (like half as many events as naturals) for infinite > sets when that real range is considered constant. It is part of a dreamworld in which TO dreams that he is the new Cantor. === Subject: Re: An uncountable countable set > >> For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >> TOE-Knee What is this IFR, inverse function rule? I've heard you mention it. > Is it just general EF? >Ross > Hey Ross! The Inverse Function Rule uses infinite-case induction to finely order infinite sets of reals mapped from a standard set, N. Where there is a bijection between N and a set S using f(n)=s, there is a mapping from S to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the bijection). The size of the set S over the interval [a,b] is given by finite sets of reals. The number of square roots, for instance, between 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to sqrt(10000). IFR can easily be used to show that the evens are half as numerous as the naturals, and other interesting facts. EF is the special case of IFR mapping the naturals in [0,oo) to the reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you define the equivalency function? Given this mapping, we can say g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in each unit interval, over oo unit intervals. Does that sound about right? Tony === Subject: Re: An uncountable countable set > >> For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >> TOE-Knee What is this IFR, inverse function rule? I've heard you mention it. > Is it just general EF? >Ross >Hey Ross! >The Inverse Function Rule uses infinite-case induction to finely order > infinite sets of reals mapped from a standard set, N. It is merely another delusion of TO's that such a rule means anything to anyone except TO. === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> <45422275@news2.lightlink The Inverse Function Rule uses infinite-case induction to finely order > infinite sets of reals mapped from a standard set, N. Where there is a > bijection between N and a set S using f(n)=s, there is a mapping from S > to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the > bijection). The size of the set S over the interval [a,b] is given by > finite sets of reals. The number of square roots, for instance, between > 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to > sqrt(10000). IFR can easily be used to show that the evens are half as > numerous as the naturals, and other interesting facts. EF is the special case of IFR mapping the naturals in [0,oo) to the > reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you > define the equivalency function? Given this mapping, we can say > g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in > each unit interval, over oo unit intervals. Does that sound about right? Tony Isn't there symmetry about the origin thus it's 2 times oo^2? Obviously half of the integers are even. What are cases against use or validity of IFR? How do you address those? Ross === Subject: Re: An uncountable countable set >> The Inverse Function Rule uses infinite-case induction to finely order >> infinite sets of reals mapped from a standard set, N. Where there is a >> bijection between N and a set S using f(n)=s, there is a mapping from S >> to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the >> bijection). The size of the set S over the interval [a,b] is given by >> finite sets of reals. The number of square roots, for instance, between >> 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to >> sqrt(10000). IFR can easily be used to show that the evens are half as >> numerous as the naturals, and other interesting facts. >> EF is the special case of IFR mapping the naturals in [0,oo) to the >> reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you >> define the equivalency function? Given this mapping, we can say >> g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in >> each unit interval, over oo unit intervals. Does that sound about right? >> Tony >Isn't there symmetry about the origin thus it's 2 times oo^2? Oh sure, it could be. I generally think of N as the positive side of the real line, so that would be right, but one could use N as both sides. As long as one chooses a standard infinite length, IFR works over the real line. >Obviously half of the integers are even. I think that's obvious. >What are cases against use or validity of IFR? How do you address > those? >Ross > Oh, there have been plenty of objections. Foremost, I think, is its dependence on infinite-case induction, hence Chas' diagonal staircase counterexample for such induction. But, I haven't seen a valid counterexample yet, though that was a good try on Chas' part. Of course, it leads to the notion of infinitesimals like EF does - that seems to be out of vogue this century, and gets plenty of objections. Virgil complained at one point that not every function has an inverse function, but any set of reals bijected with another does so with an inherently invertible function, or there's no bijection. Also it only applies to real values, since it depends on a value range, and the infinite range of the line is not considered real by standard mathematicians. I don't care - call it N, and express sets over the real range as formulas on N, or Big'un. It works for finite and infinite sets of real values mapped from the naturals. Do you see any problems with it? You might want to play with it. Maybe you'll extend it somehow. I certainly was happy to see it mesh with your EF. I had been trying to apply it to the relationship between the continuum and the naturals using f(x)=log_oo(x) or something, but that caused problems. EF is the correct mapping between the two, according to IFR. :) Rock on! Tony === Subject: Re: An uncountable countable set >> Please identify something you see as incorrect or don't understand. > What's the point? People have been pointing out your incorrect > statements for years. You just sail right past every time. >> So really think you are correct and that you make sense? >> It must be nice. >> MoeBlee > For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. >I think this controversy is in fact the difference between relying > utterly on inutition (as Ross seems to do) and relying utterly on > logical conclusions of some explicit set of assumptions (which is the > domain of what I would call mathematics) which is at issue here. >Just as logical conclusions from some set of assumptions can at times > be in conflict with our intuitions; so it is also true that we can hold > intuitions which are not logically compatible with each other. >In my use of the word, mathematics is within that domain of discussion > which eschews the latter in favor of the former. It is a specialisation > of the domain of logic, rather than of the domain of physics. > >> Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. >Of course; there is nothing he says that is completely without /some/ > sort of sense. But I would say he is speaking /poetically/, not > mathematically; so in the context of sci.math, I can't respond to his > remarks. >(So if you read this Ross; it's not that I don't respond to you because > I don't like you; I don't respond to you becuase I have no common > conceptual gound with you. You actually strike me as a pretty nice guy, > overalll. You're always polite and well meaning; that's all one can > ask!) >A poet would say that A rose is still a rose by any other name; a > mathematician would say that By 'a rose' we mean a repesentative of an > equivalence class of those herbacious plants having the following > properties: thorns, leaves found on alternating sides of the stem; > we can deduce that the assertion of the heavy metal ballad, 'Every rose > has its thorn', logically follows. >Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. Very true, but like the Zen archer, we have to train our intuitions, and when they are in harmony with the universe, the arrow hits its mark.:) > >> But, of course, >> coming from me, that probably doesn't mean much. :) >On the contrary; I think you have accurately identified the nub of > whatever controversy has arisen regarding arguments you have asserted > in this and any other threads. > I've ended up screwing my brow trying to figure out what the crux of the disagreement was. It's not always easy to dig down to the logical roots and pinpoint the exact problem that leads to differing conclusions. Of course, I rather doubt that means you agree with all my points, but that okay. Agreeing all the time gets a little boring, eh? Have a nice day. Tony === Subject: Re: An uncountable countable set > Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. >Very true, but like the Zen archer, we have to train our intuitions, and > when they are in harmony with the universe, the arrow hits its mark.:) TO's intuition is so far out of harmony with standard mathematics that when he looses his arrows, they behave like boomerangs and hit him in the ass. === Subject: Re: An uncountable countable set >Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. >> Very true, but like the Zen archer, we have to train our intuitions, and >> when they are in harmony with the universe, the arrow hits its mark.:) >TO's intuition is so far out of harmony with standard mathematics that > when he looses his arrows, they behave like boomerangs and hit him in > the ass. Did you notice that I catch them every time, without wincing? === Subject: Re: An uncountable countable set >Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. >> Very true, but like the Zen archer, we have to train our intuitions, and >> when they are in harmony with the universe, the arrow hits its mark.:) >TO's intuition is so far out of harmony with standard mathematics that > when he looses his arrows, they behave like boomerangs and hit him in > the ass. >Did you notice that I catch them every time, without wincing? Always knew TO was a hardass. === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> <45421ea8@news2.lightlink> A poet would say that A rose is still a rose by any other name; a > mathematician would say that By 'a rose' we mean a repesentative of an > equivalence class of those herbacious plants having the following > properties: thorns, leaves found on alternating sides of the stem; > we can deduce that the assertion of the heavy metal ballad, 'Every rose > has its thorn', logically follows. > Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. Very true, but like the Zen archer, we have to train our intuitions, and > when they are in harmony with the universe, the arrow hits its mark.:) > In physics those intuitions are the ones which accord with the universe of real world measurements. In mathematics, those intuitions are the ones which are instead in accord with the universe of logical conclusions from agreed upon premises. The archers are aiming at different targets; so they develop different intuitions. === Subject: Re: An uncountable countable set > A poet would say that A rose is still a rose by any other name; a > mathematician would say that By 'a rose' we mean a repesentative of an > equivalence class of those herbacious plants having the following > properties: thorns, leaves found on alternating sides of the stem; > we can deduce that the assertion of the heavy metal ballad, 'Every rose > has its thorn', logically follows. >> Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. >> Very true, but like the Zen archer, we have to train our intuitions, and >> when they are in harmony with the universe, the arrow hits its mark.:) >In physics those intuitions are the ones which accord with the > universe of real world measurements. In mathematics, those intuitions > are the ones which are instead in accord with the universe of logical > conclusions from agreed upon premises. >The archers are aiming at different targets; so they develop different > intuitions. > That's very poetic, and as such, I don't find it mathematically compelling. The universe is the product of numbers. :) TOEknee === Subject: Re: An uncountable countable set > The universe is the product of numbers. :) >TOEknee LEGparts is wrong! As Usual!! The universe might as easily be a sum of numbers quite as being a product of them, but seems to involve things which are not prurely numbers, too. === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) Coming from TO it damns Ross. Even by your standards, Virgil, this is egregiously silly. TO skips the basic exposition in Robinson's book, but finds a sentence he likes. So this damns Robinson's non-standard analysis, does it? Brian Chandler http://imaginatorium.org === Subject: Re: An uncountable countable set > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) >> Coming from TO it damns Ross. >Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? >Brian Chandler > http://imaginatorium.org > No, he's saying my endorsement of Ross Finlayson's ideas reduces his credibility, or some such. I didn't skip the basic exposition. I skipped ahead from where he was talking about stratified sentences, leaving a bookmark there to return, because it was very heady and technical. Have you read it? You should. Then talk. I have not found one sentence I like, but conclusion after conclusion all in line with what I've been saying, though I questioned one of his conclusions, without disagreeing. In all, he makes perfect sense so far. In any case, if you're going to respond to Virgil's silliest comments (I didn't bother with this one), you might want to pay attention to what he's being silly about. TOny === Subject: Re: An uncountable countable set >For what it's worth, and I know this doesn't add a lot of credibility to >Ross in your eyes, coming from me, but I think Ross has a genuine >intuition that isn't far off with respect to what's controversial in >modern math. Sure, he gets repetitive and I don't agree with everything >he says, but his cryptic Well order the reals, which I actually >haven't seen too much of lately, is a direct reference to his EF >(Equivalence Function, yes?) between the naturals and the reals in >[0,1). The reals viewed as discrete infinitesimals map to the >hypernaturals, anyway, and his EF is a special case of my IFR. So, to >answer your question, I think Ross makes some sense. But, of course, >coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. >Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? You are mixing up Ross A. Finlayson with Abraham Robinson. I have nothing against Robinson or his non-standard analysis, in fact, I rather like it. === Subject: Re: An uncountable countable set For what it's worth, and I know this doesn't add a lot of credibility to >Ross in your eyes, coming from me, but I think Ross has a genuine >intuition that isn't far off with respect to what's controversial in >modern math. Sure, he gets repetitive and I don't agree with everything >he says, but his cryptic Well order the reals, which I actually >haven't seen too much of lately, is a direct reference to his EF >(Equivalence Function, yes?) between the naturals and the reals in >[0,1). The reals viewed as discrete infinitesimals map to the >hypernaturals, anyway, and his EF is a special case of my IFR. So, to >answer your question, I think Ross makes some sense. But, of course, >coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. >Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? Virgil said Ross, not Robinson, I believe. -- David Marcus === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> For what it's worth, and I know this doesn't add a lot of credibility to >Ross in your eyes, coming from me, but I think Ross has a genuine >intuition that isn't far off with respect to what's controversial in >modern math. Sure, he gets repetitive and I don't agree with everything >he says, but his cryptic Well order the reals, which I actually >haven't seen too much of lately, is a direct reference to his EF >(Equivalence Function, yes?) between the naturals and the reals in >[0,1). The reals viewed as discrete infinitesimals map to the >hypernaturals, anyway, and his EF is a special case of my IFR. So, to >answer your question, I think Ross makes some sense. But, of course, >coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? Virgil said Ross, not Robinson, I believe. Yes, of course. But Virgil's implication is that TO says person P is right about something implies P is wrong. This may, contingently, be true about Ross, but the argument could equally be applied to Robinson, in which case the conclusion is obviously not true. Brian Chandler http://imaginatorium.org === Subject: Re: An uncountable countable set > > For what it's worth, and I know this doesn't add a lot of credibility >> to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with >> everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >Coming from TO it damns Ross. > Even by your standards, Virgil, this is egregiously silly. TO skips the >basic exposition in Robinson's book, but finds a sentence he likes. So >this damns Robinson's non-standard analysis, does it? > Virgil said Ross, not Robinson, I believe. >Yes, of course. But Virgil's implication is that TO says person P is > right about something implies P is wrong. This may, contingently, be > true about Ross, but the argument could equally be applied to Robinson, > in which case the conclusion is obviously not true. Since Ross has, by his own posts, shown himself to be as far out of touch with reality as TO, TO's approval is only piling Pelion on Ossa. If TO were to support someone reasonably in touch with mathematical reality, I should not have regarded it as a last straw situation. === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> .. Since Ross has, by his own posts, shown himself to be as far out of > touch with reality as TO, TO's approval is only piling Pelion on Ossa. If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. Virgil's among the few people here ever called a liar. His overgeneralizations are generally wrong. Is that like the one about wrestling with a pig? Virgil, are you considering attempting to converse directly with me? Last time that happened I showed you were wrong, no? You piped up with one of your little barbs and ate it. Virgil, I don't care for what you say and don't think you have anything else to tell me, so, don't, punk. Virgil, you're snotnosed and rude, fraud. Is it a bitter pill, Virgil? I've found mistakes in the CRC, Knuth, Mathworld, maybe HMF, etc. To paraphrase Fraenkel: reliance on transfinite cardinals is a mistake. Measure theory is about quantitative continua, there are no applications of transfinite cardinals, and the integral calculus is a nilpotent infinitesimal analysis. Re-Vitali-ize measure theory, polydimensionally. Hilbert requests a well-ordering of the reals. There is no universe in ZFC. The only theory with no axioms is A theory. The Finlayson numbers are all the numbers, the Finlayson reals are the reals. Select. Ross === Subject: Re: An uncountable countable set > >> For what it's worth, and I know this doesn't add a lot of credibility >> to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with >> everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. >> Even by your standards, Virgil, this is egregiously silly. TO skips the >> basic exposition in Robinson's book, but finds a sentence he likes. So >> this damns Robinson's non-standard analysis, does it? > Virgil said Ross, not Robinson, I believe. >> Yes, of course. But Virgil's implication is that TO says person P is >> right about something implies P is wrong. This may, contingently, be >> true about Ross, but the argument could equally be applied to Robinson, >> in which case the conclusion is obviously not true. > > Since Ross has, by his own posts, shown himself to be as far out of > touch with reality as TO, TO's approval is only piling Pelion on Ossa. >If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. Like Boole? === Subject: Re: An uncountable countable set > If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. >Like Boole? Boole does not need, or want, TO's support. === Subject: Re: An uncountable countable set > >> For what it's worth, and I know this doesn't add a lot of credibility >> to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with >> everything >> he says, but his cryptic Well order the reals, which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. >> Even by your standards, Virgil, this is egregiously silly. TO skips the >> basic exposition in Robinson's book, but finds a sentence he likes. So >> this damns Robinson's non-standard analysis, does it? > Virgil said Ross, not Robinson, I believe. >> Yes, of course. But Virgil's implication is that TO says person P is >> right about something implies P is wrong. This may, contingently, be >> true about Ross, but the argument could equally be applied to Robinson, >> in which case the conclusion is obviously not true. > > Since Ross has, by his own posts, shown himself to be as far out of > touch with reality as TO, TO's approval is only piling Pelion on Ossa. >If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. Like Robinson? === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> <4542a3a5@news2.lightlink > If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. Like Robinson? Oy vey. Orlow mistakes letting his eyes roam over the pages of an advanced text in mathematics for understanding that mathematics and giving it his moral support. MoeBlee === Subject: Re: An uncountable countable set > If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a last straw situation. >Like Robinson? Robinson does not need, or want, TO's support. === Subject: Re: An uncountable countable set >> For what it's worth, and I know this doesn't add a lot of credibility to >>Ross in your eyes, coming from me, but I think Ross has a genuine >>intuition that isn't far off with respect to what's controversial in >>modern math. Sure, he gets repetitive and I don't agree with everything >>he says, but his cryptic Well order the reals, which I actually >>haven't seen too much of lately, is a direct reference to his EF >>(Equivalence Function, yes?) between the naturals and the reals in >>[0,1). The reals viewed as discrete infinitesimals map to the >>hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>answer your question, I think Ross makes some sense. But, of course, >>coming from me, that probably doesn't mean much. :) > Coming from TO it damns Ross. >>Even by your standards, Virgil, this is egregiously silly. TO skips the >> basic exposition in Robinson's book, but finds a sentence he likes. So >> this damns Robinson's non-standard analysis, does it? >> Virgil said Ross, not Robinson, I believe. Yes, of course. But Virgil's implication is that TO says person P is >right about something implies P is wrong. This may, contingently, be >true about Ross, but the argument could equally be applied to Robinson, >in which case the conclusion is obviously not true. Well technically, Brian, you're being reasonable for a change. ~v~~ === Subject: Re: An uncountable countable set > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) >> Coming from TO it damns Ross. > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? >> Virgil said Ross, not Robinson, I believe. >Yes, of course. But Virgil's implication is that TO says person P is > right about something implies P is wrong. This may, contingently, be > true about Ross, but the argument could equally be applied to Robinson, > in which case the conclusion is obviously not true. >Brian Chandler > http://imaginatorium.org > And, what about those rare occasions when I agree with Virgil? Uh oh. === Subject: Re: An uncountable countable set > > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) >> Coming from TO it damns Ross. > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? >> Virgil said Ross, not Robinson, I believe. >Yes, of course. But Virgil's implication is that TO says person P is > right about something implies P is wrong. This may, contingently, be > true about Ross, but the argument could equally be applied to Robinson, > in which case the conclusion is obviously not true. >Brian Chandler > http://imaginatorium.org And, what about those rare occasions when I agree with Virgil? Uh oh. It actually has happened that TO and I agree on something. It does not happen boringly often but it does happen. === Subject: Re: An uncountable countable set > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic Well order the reals, which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) >> Coming from TO it damns Ross. > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this damns Robinson's non-standard analysis, does it? >> Virgil said Ross, not Robinson, I believe. > Yes, of course. But Virgil's implication is that TO says person P is > right about something implies P is wrong. This may, contingently, be > true about Ross, but the argument could equally be applied to Robinson, > in which case the conclusion is obviously not true. >> Brian Chandler > http://imaginatorium.org > And, what about those rare occasions when I agree with Virgil? Uh oh. >It actually has happened that TO and I agree on something. >It does not happen boringly often but it does happen. Does that mean that my assent discredits anything that you have to say? I don't think you see it that way. You sometimes make some very good points, although usually there are an almost uncountably many slights and poobars interspersed. === Subject: Re: An uncountable countable set >> And, what about those rare occasions when I agree with Virgil? Uh oh. >It actually has happened that TO and I agree on something. >It does not happen boringly often but it does happen. >Does that mean that my assent discredits anything that you have to say? No! TO is not wrong reliably enough so that one may always assume the opposite of what he claims. === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> When mathematicians talk, they know which words have technical meanings > and which don't. Ross simply uses the words without knowing the > technical meanings. So, it gives the appearance of mathematics, but > there is no actual communication of mathematical ideas. I once tried to teach some mathematics to some friends who weren't > mathematicians via a weekly lunch seminar. We took a (fairly advanced) > math book, and started reading it. When I read a math book, I can > immediately categorize each sentence as definition, theorem, proof, or > remark, even if the sentence isn't labeled as such. I was a bit > surprised to discover that my friends weren't picking up on this at all. > As such, they couldn't even begin to follow the book, since they > couldn't tell what the purpose of each sentence was in the logical flow. > They weren't even aware of the convention that a word in italics means > the sentence is a definition of the word. -- > I dispute that. Let's see, the nearest mathematics book is a Dover reprint of Meyer's _Introduction to Mathematical Fluid Dynamics_. So, explain fluid mechanics. Perhaps you feel more secure in your grasp of the subject talking about set theory? So do I. Ross === Subject: Re: An uncountable countable set > When mathematicians talk, they know which words have technical meanings > and which don't. Ross simply uses the words without knowing the > technical meanings. So, it gives the appearance of mathematics, but > there is no actual communication of mathematical ideas. > I once tried to teach some mathematics to some friends who weren't > mathematicians via a weekly lunch seminar. We took a (fairly advanced) > math book, and started reading it. When I read a math book, I can > immediately categorize each sentence as definition, theorem, proof, or > remark, even if the sentence isn't labeled as such. I was a bit > surprised to discover that my friends weren't picking up on this at all. > As such, they couldn't even begin to follow the book, since they > couldn't tell what the purpose of each sentence was in the logical flow. > They weren't even aware of the convention that a word in italics means > the sentence is a definition of the word. >I dispute that. You dispute what, exactly? > Let's see, the nearest mathematics book is a Dover reprint of Meyer's > _Introduction to Mathematical Fluid Dynamics_. So, explain fluid > mechanics. >Perhaps you feel more secure in your grasp of the subject talking about > set theory? So do I. -- David Marcus === Subject: Re: An uncountable countable set <45417528$1@news2.lightlink> Coming from TO it damns Ross. Virgil, shut up. ZFC and IST, Nelson's Internal Set theory, and Robinson's hyperreals are quite compatible with each other, for example how ZFC is coconsistent with IST. ZF is inconsistent. There is no set of sets in ZF. Robinson's hyperreals are basically what are referred to here as Newton's notion of the reals, with fluents and fluxions, extended with infinite values. The hyperintegers are similarly a notion of the finite natural numbers having appended infinite, natural numbers. Fluxions after the first unit's aren't generally used in analysis, but they can be in chaining derivatives, composition, as Newton's are nilpotent. The sum of any number of zeros is zero. There are quite the few other even more alternative and nonstandard formulations of the real numbers than the hyperreals, so constructed in and known from the literature, re Schmieden and Laugwitz, Bishop and Cheng, myself, etcetera. There are as well useful compactifications or projective extensions of the real numbers with the as you say point at infinity. Ross === Subject: Re: An uncountable countable set >> t=0 is precluded by n e N and t(n) = -1/n. > Really? >> I hope you will accept as true that noon occurred yesterday. >> Let's define noon yesterday as t=0. Now let's define a set of values > t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE > natural numbers n. >> Has my giving these names to those times somehow > precluded noon yesterday from occurring? Retroactively? > Do you live in the gedanken? Oy. Nothing happens at noon. >Did noon occur? Not within the constraints of the experiment. Nothing is allowed to happen at noon. > >> Your desired result does not happen before noon. >What desired result? I didn't have an experiment, I > just named a bunch of times. Is noon precluded by > my defining that countable set of variables? > > Can anything happen at noon? What can change, in the vase, at noon? What is the state before noon? === Subject: Re: An uncountable countable set yet more analysis that just blows me away with its mathematical and >rhetorical brilliance: Well obviously it blows you away with its rhetorical brilliance or you wouldn't bother to front post. As to its mathmatical significance I'll grant you its brilliance. As to its jazz significance I couldn't be bothered to say. > So your arguments > for establishment views >>I am interested in learning ZF and related theories, but I do not claim >>that ZF is superior to any other theory. >> ZF and its kin are not theories because they can't be proven true. >> They're only analytical methods. >> In my arguments with cranks >> Ah yes the operative mathematically exhaustive definition for crank >> being crank(x)=disagree(u). >> I >>do not claim that ZF is the best theory (I have no opinion really on a >>best theory, especially since at this stage I am just trying to >>learn, not make definitive normative judgments about best), >> Which is certainly wise when you have no idea whether your analytical >> method is actually true. >> but >>rather my points usually are to correct mistatements about what the >>actual formulations and theorems of set theory are, and to point out >>that the crank alternatives are not formal theories and not even within >>a thousand miles of being amenable to being a formal or EVEN coherent >>theory. >> Yes well you've certainly pointed that out on numerous occasions with >> your own opinions as to what's what and when requested have even been >> polite enough to back up those opinions with other opinions and when >> all else failed have demanded people with differing opinions research >> your opinions and document them for you. So what? Your virtue for what >> it's worth is completely intact. > It's all just a smoke > screen as far as I'm concerned, so much jargon and verbiage used to > simulate a sophisticated technical mathematical edifice where there is > none. >>And, since I have studied the subject, I know that you are wrong on >>that matter. The systems and terminology are given precisely (or can >>easily be made PERFECTLY precise with a bit of work on the reader's >>part, if perfect precision is required, as I do happen to require it of >>set theory). >> Yadayada whatever, Moe. More opinions to back up other opinions >> followed by demands others research and justify your opinions for you. >>You can claim a smoke screen all you want, but you won't even look at >>the actual performance of the mathematics you call a smoke screen. So >>of course, there is no possibility of convincing you that the system >>and terminology is precise. >> I never said it wasn't precise, Moe, I said precision didn't matter >> when the issue was truth and the exhaustion of truth instead. What the >> hell would the precision of pi mean if you're dealing with straight >> lines instead of circular arcs. Precision is irrelevant if your >> analytical technique is incorrect to begin with. >> I could tell you that there exists a >>portable hard drive that carries more than a gigabyte in a device >>smaller than a pack of gum, and you can say forever that no such thing >>exists, since you won't even look at a showing of such a device. >> I'm only interested in your opinions to the extent the truth of your >> opinions is demonstrable. You're opinions on math don't fall into the >> same category because they aren't, you can't, and you're perfectly >> content with the assumption of truth instead. That's the difference. >> I can >>say that set theory is a completely rigorous system that axiomatizes >>the usual theorems of real analysis, and you can say forever that it is >>not rigorous, just a smoke screen, forever, since you won't read a book >>that shows just such a rigorous axiomatization. >> You mean I won't research and justify your opinions for you on your >> sayso when you can't even begin to do it for yourself? Why bother? ~v~~ === Subject: Re: An uncountable countable set <452e8ea4@news2.lightlink> <4531023e@news2.lightlink> <45341984@news2.lightlink> <45359256@news2.lightlink> <4539125e$1@news2.lightlink> <453d9a7c$1@news2.lightlink> <453e4a85@news2.lightlink> <453e824b@news2.lightlink> <453fb285@news2.lightlink > Your examples of the circle and rectangle are good. Neither has a > height > outside of its x range. The height of the circle is 0 at x=-1 and x=1, > because the circle actually exists there. To ask about its height at > x=9 > is like asking how the air quality was on the 85th floor of the World > Trade Center yesterday. Similarly, it makes little sense to ask what > happens at noon. There is no vase at noon. >> Do you really mean to say that there is no vase at noon or do you mean >> to say that the vase is not empty at noon? > If noon exists at all, the vase is not empty. All finite naturals will > have been removed, but an infinite number of infinitely-numbered balls > will remain. >> If noon exists at all? How do we decide? > We decide on the basis of whether 1/n=0. Is that possible for n in N? > Hmmmm......nope. >> So, noon doesn't exist. And, there is no vase at noon. I thought you >> were saying the vase contains an infinite number of balls at noon. > If the vase exists at noon, then it has an uncountable number of balls > labeled with infinite values. But, no infinite values are allowed i the > experiment, so this cannot happen, and noon is excluded. >> So did the North Koreans nuke the vase before noon? >> The only relevant issue is whether according to the rules set up in the >> problem, is each ball inserted before noon also removed before noon? >> An affirmative confirms that the vase is empty at noon. >> A negative directly violates the conditions of the problem. >> How does TO answer? > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. >> Your response requires that the vase contains balls which were >> never, by the stated rules, put in. >> You keep saying things like if the clock runs till noon there are >> balls with infinite numbers on them even though the rules say there >> are >> no balls with infinite numbers on them. How do you reconcile that? >> If I put in balls 1, 2, 3 and stop, can the clock tick till noon >> without >> requiring a 4th ball? >> If I specify times for balls 1-1000 only, can the clock till noon >> without >> requiring a 1001-th ball? >> How is it, in your world, that when I specify times for all natural >> numbered >> balls, I am required to put in balls that don't have natural numbers? > The problem is that Tony thinks time is a function of the number of > insertions you've gone through. In order to get to any particular > time you have to perform the insertions up to that point. He then > thinks that if you want to get to noon, you have to have performed > some infinite (whatever that means) iterations, where balls without > natural numbers are inserted. That this is obviously not what the > problem statement says doesn't seem to bother him. Nor that it's > absolutely nothing like an intuitive picture of what time is. Time is ultimately irrelevant in this gedanken, but if it is to be > considered, the constraints regarding time cannot be ignored. Is time relevent to the question or isn't it? If it isn't, why must these constraints be respected? > Events occurring in time must occupy at least one moment. I have no idea what this is supposed to mean. > Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. No, you have the concept of the independent variable bent. No, you have the concept of the independent variable bent. > The number of balls is related to the time by a formula which works in both directions. For any iteration in the sequence of insertions/removals you can work out what time it occurs at if you know what number the iteration is indexed by. This doesn't imply that noon does not exist unless there is an iteration that corresponds to it. That is a complete non sequitur and, I think, the root logical error that you make. > So, when does the vase become empty? Nothing can occur at noon, as far > as ball removals. AT every time before noon, balls are in the vase. So, > when does the vase become empty At every time before noon (after 1 minute to noon) there are balls in the vase. At noon, there are no balls in the vase. So I guess one would say the vase becomes empty at noon. >, and how? By every ball that was inserted having been removed. Now correct me if I'm wrong, but I think you agreed that every specific ball has been removed before noon. And indeed the problem statement doesn't mention any non-specific balls, so it seems that the vase must be empty. However, you believe that in order to reach noon one must have iterations where non specific balls without natural numbers are inserted into the vase and thus, if the problem makes sense and noon is meaningful, the vase is non-empty at noon. Is this a fair summary of your position? If so, I'd like to make clear that I have no idea in the world why you hold such a notion. It seems utterly illogical to me and it baffles me why you hold to it so doggedly. So, I'd like to try and understand why you think that it is the case. If you can explain it cogently, maybe I'll be convinced that you make sense. And maybe if you can't explain, you'll admit that you might be wrong? Let's start simply so there is less room for mutual incomprehension. Let's imagine a new experiment. In this experiment, we have the same infinite vase and the same infinite set of balls with natural numbers on them. Let's call the time one minute to noon -1 and noon 0. Note that time is a real-valued variable that can have any real value. At time -1/n we insert ball n into the vase. My question : what do you think is in the vase at noon? -- mike. === Subject: Re: An uncountable countable set > Your examples of the circle and rectangle are good. Neither has a > height > outside of its x range. The height of the circle is 0 at x=-1 and x=1, > because the circle actually exists there. To ask about its height at > x=9 > is like asking how the air quality was on the 85th floor of the World > Trade Center yesterday. Similarly, it makes little sense to ask what > happens at noon. There is no vase at noon. >> Do you really mean to say that there is no vase at noon or do you mean >> to say that the vase is not empty at noon? > If noon exists at all, the vase is not empty. All finite naturals will > have been removed, but an infinite number of infinitely-numbered balls > will remain. >> If noon exists at all? How do we decide? > We decide on the basis of whether 1/n=0. Is that possible for n in N? > Hmmmm......nope. >> So, noon doesn't exist. And, there is no vase at noon. I thought you >> were saying the vase contains an infinite number of balls at noon. > If the vase exists at noon, then it has an uncountable number of balls > labeled with infinite values. But, no infinite values are allowed i the > experiment, so this cannot happen, and noon is excluded. >> So did the North Koreans nuke the vase before noon? >> The only relevant issue is whether according to the rules set up in the >> problem, is each ball inserted before noon also removed before noon? >> An affirmative confirms that the vase is empty at noon. >> A negative directly violates the conditions of the problem. >> How does TO answer? > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. >> Your response requires that the vase contains balls which were >> never, by the stated rules, put in. >> You keep saying things like if the clock runs till noon there are >> balls with infinite numbers on them even though the rules say there >> are >> no balls with infinite numbers on them. How do you reconcile that? >> If I put in balls 1, 2, 3 and stop, can the clock tick till noon >> without >> requiring a 4th ball? >> If I specify times for balls 1-1000 only, can the clock till noon >> without >> requiring a 1001-th ball? >> How is it, in your world, that when I specify times for all natural >> numbered >> balls, I am required to put in balls that don't have natural numbers? > The problem is that Tony thinks time is a function of the number of > insertions you've gone through. In order to get to any particular > time you have to perform the insertions up to that point. He then > thinks that if you want to get to noon, you have to have performed > some infinite (whatever that means) iterations, where balls without > natural numbers are inserted. That this is obviously not what the > problem statement says doesn't seem to bother him. Nor that it's > absolutely nothing like an intuitive picture of what time is. >> Time is ultimately irrelevant in this gedanken, but if it is to be >> considered, the constraints regarding time cannot be ignored. >Is time relevent to the question or isn't it? If it isn't, why must > these constraints be respected? > >> Events occurring in time must occupy at least one moment. >I have no idea what this is supposed to mean. >Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. >> No, you have the concept of the independent variable bent. >No, you have the concept of the independent variable bent. > >> The number of balls is related to the time by a formula which works in both directions. >For any iteration in the sequence of insertions/removals you can work > out what time it occurs at if you know what number the iteration is > indexed by. This doesn't imply that noon does not exist unless there > is an iteration that corresponds to it. That is a complete non sequitur > and, I think, the root logical error that you make. > >> So, when does the vase become empty? Nothing can occur at noon, as far >> as ball removals. AT every time before noon, balls are in the vase. So, >> when does the vase become empty >At every time before noon (after 1 minute to noon) there are balls in > the vase. At noon, there are no balls in the vase. So I guess one would > say the vase becomes empty at noon. > >> , and how? >By every ball that was inserted having been removed. >Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >My question : what do you think is in the vase at noon? > A countable infinity of balls. This is very simple. Everything that occurs is either an addition of ten balls or a removal of 1, and occurs a finite amount of time before noon. At the time of each event, balls remain. At noon, no balls are inserted or removed. The vase can only become empty through the removal of balls, so if no balls are removed, the vase cannot become empty at noon. It was not empty before noon, therefore it is not empty at noon. Nothing can happen at noon, since that would involve a ball n such that 1/n=0. === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. >So, noon exists in this case, even though nothing happens at noon. Not really, but there is a big difference between this and the original experiment. If noon did exist here as the time of any event (insertion), then you would have an UNcountably infinite set of balls. Presumably, given only naturals, such that nothing is inserted at noon, by noon all naturals have been inserted, for the countable infinity. Then insertions stop, and the vase has what it has. The issue with the original problem is that, if it empties, it has to have done it before noon, because nothing happens at noon. You conclude there is a change of state when nothing happens. I conclude there is not. > >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. The vase can only become empty through the removal of balls, >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > === Subject: Re: An uncountable countable set > So, noon exists in this case, even though nothing happens at noon. >Not really Yes really. === Subject: Re: An uncountable countable set So, noon exists in this case, even though nothing happens at noon. >> Not really >Yes really. Nope. (see? I can be an imbecile too) === Subject: Re: An uncountable countable set So, noon exists in this case, even though nothing happens at noon. >> Not really >Yes really. >Nope. > (see? I can be an imbecile too) How can there be times before noon without a noon? What time is some undefined time minus one minute? TO claims to know. === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. >So, noon exists in this case, even though nothing happens at noon. >Not really, but there is a big difference between this and the original > experiment. If noon did exist here as the time of any event (insertion), > then you would have an UNcountably infinite set of balls. Presumably, > given only naturals, such that nothing is inserted at noon, by noon all > naturals have been inserted, for the countable infinity. Then insertions > stop, and the vase has what it has. The issue with the original problem > is that, if it empties, it has to have done it before noon, because > nothing happens at noon. You conclude there is a change of state when > nothing happens. I conclude there is not. So, noon doesn't exist in this case either? -- David Marcus === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. > So, noon exists in this case, even though nothing happens at noon. >> Not really, but there is a big difference between this and the original >> experiment. If noon did exist here as the time of any event (insertion), >> then you would have an UNcountably infinite set of balls. Presumably, >> given only naturals, such that nothing is inserted at noon, by noon all >> naturals have been inserted, for the countable infinity. Then insertions >> stop, and the vase has what it has. The issue with the original problem >> is that, if it empties, it has to have done it before noon, because >> nothing happens at noon. You conclude there is a change of state when >> nothing happens. I conclude there is not. >So, noon doesn't exist in this case either? > Nothing happens at noon, and as long as there is no claim that anything happens at noon, then there is no problem. Before noon there was an unboundedly large but finite number of balls. At noon, it is the same. === Subject: Re: An uncountable countable set > So, noon doesn't exist in this case either? Nothing happens at noon, and as long as there is no claim that anything > happens at noon, then there is no problem. By noon, it is all over and every ball has been removed. >Before noon there was an > unboundedly large but finite number of balls. At noon, it is the same. Let's see! The first numbered ball is removed before noon. If the nth ball is removed before noon, then so is the n+1st ball. So that in any system consistent with ZF or NBG, EVERY naturally numbered ball (which is all the balls allowed to be inserted by the gedankenexperiment) are removed. === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. > So, noon exists in this case, even though nothing happens at noon. >> Not really, but there is a big difference between this and the original >> experiment. If noon did exist here as the time of any event (insertion), >> then you would have an UNcountably infinite set of balls. Presumably, >> given only naturals, such that nothing is inserted at noon, by noon all >> naturals have been inserted, for the countable infinity. Then insertions >> stop, and the vase has what it has. The issue with the original problem >> is that, if it empties, it has to have done it before noon, because >> nothing happens at noon. You conclude there is a change of state when >> nothing happens. I conclude there is not. >So, noon doesn't exist in this case either? >Nothing happens at noon, and as long as there is no claim that anything > happens at noon, then there is no problem. Before noon there was an > unboundedly large but finite number of balls. At noon, it is the same. So, noon does exist in this case? -- David Marcus === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. > So, noon exists in this case, even though nothing happens at noon. >> Not really, but there is a big difference between this and the original >> experiment. If noon did exist here as the time of any event (insertion), >> then you would have an UNcountably infinite set of balls. Presumably, >> given only naturals, such that nothing is inserted at noon, by noon all >> naturals have been inserted, for the countable infinity. Then insertions >> stop, and the vase has what it has. The issue with the original problem >> is that, if it empties, it has to have done it before noon, because >> nothing happens at noon. You conclude there is a change of state when >> nothing happens. I conclude there is not. > So, noon doesn't exist in this case either? >> Nothing happens at noon, and as long as there is no claim that anything >> happens at noon, then there is no problem. Before noon there was an >> unboundedly large but finite number of balls. At noon, it is the same. >So, noon does exist in this case? > Since the existence of noon does not require any further events, it's a moot point. As I think about it, no, noon does not exist in this problem either, as the time of any event, since nothing is removed at noon. It is also not required for any conclusion, except perhaps that there are uncountably many balls, rather than only countably many. But, there are only countably many balls, so, no, noon is not part of the problem here. As we approach noon, the limit is 0. We don't reach noon. === Subject: Re: An uncountable countable set > Now correct me if I'm wrong, but I think you agreed that every > specific ball has been removed before noon. And indeed the problem > statement doesn't mention any non-specific balls, so it seems that > the vase must be empty. However, you believe that in order to reach > noon one must have iterations where non specific balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and noon is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? >> If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cogently, maybe > I'll be convinced that you make sense. And maybe if you can't explain, > you'll admit that you might be wrong? >> Let's start simply so there is less room for mutual incomprehension. > Let's imagine a new experiment. In this experiment, we have the same > infinite vase and the same infinite set of balls with natural numbers > on them. Let's call the time one minute to noon -1 and noon 0. Note > that time is a real-valued variable that can have any real value. At > time -1/n we insert ball n into the vase. >> My question : what do you think is in the vase at noon? >> A countable infinity of balls. > So, noon exists in this case, even though nothing happens at noon. >> Not really, but there is a big difference between this and the original >> experiment. If noon did exist here as the time of any event (insertion), >> then you would have an UNcountably infinite set of balls. Presumably, >> given only naturals, such that nothing is inserted at noon, by noon all >> naturals have been inserted, for the countable infinity. Then insertions >> stop, and the vase has what it has. The issue with the original problem >> is that, if it empties, it has to have done it before noon, because >> nothing happens at noon. You conclude there is a change of state when >> nothing happens. I conclude there is not. > So, noon doesn't exist in this case either? >> Nothing happens at noon, and as long as there is no claim that anything >> happens at noon, then there is no problem. Before noon there was an >> unboundedly large but finite number of balls. At noon, it is the same. >So, noon does exist in this case? >Since the existence of noon does not require any further events, it's a > moot point. As I think about it, no, noon does not exist in this problem > either, as the time of any event, since nothing is removed at noon. It > is also not required for any conclusion, except perhaps that there are > uncountably many balls, rather than only countably many. But, there are > only countably many balls, so, no, noon is not part of the problem here. > As we approach noon, the limit is 0. We don't reach noon. To recap, we add ball n at time -1/n. We don't remove any balls. With this setup, you conclude that noon does not exist. Is this correct? -- David Marcus === Subject: Re: An uncountable countable set > As we approach noon, the limit is 0. We don't reach noon. If TO never reaches noon, he must still be less than one day old. I must say he often acts like it. The rest of us manage to reach noon on the close order of once a day, with some variations for those circumnavigating the Earth at high speeds. === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. >No one disagrees with the above statements. > >> The vase can only become empty through the removal of balls, >Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >(T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > Well, you have (8), which is kind of circular, but related. >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. >Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. That is correct. Noon is incompatible with the problem statement. >Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions Right. Your conclusion is at odds with the notion that only removals may empty the vase, which seems to be an obvious assumption, no other means of achieving emptiness having been mentioned. >So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? I don't believe any of those assumptions are the problem. (2) should state that t === Subject: Re: An uncountable countable set > So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >That is correct. Noon is incompatible with the problem statement. Noon may be incompatible with TO's understanding, but without it, the original gedankenexperiment is totally undefined. > So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >I don't believe any of those assumptions are the problem. (2) should > state that t unspoken assumption on your part which causes the problem is that noon > is part of the problem. Noon may be incompatible with TO's understanding, but without it, the original gedankenexperiment is totally undefined. > Clearly, it cannot be, because anything that > happened at t=0 would involve n s.t. 1/n=t. That assumption by TO is nowhere justified by the actual statement of the gedankenexperiment. > Essentially, the problem > produces a paradox by asking a question which contradicts the situation. > Nothing happens at noon. The process never completes the unending set. Like God halting the sun in the sky so the Hebrews could keep fighting? === Subject: Re: An uncountable countable set <45341984@news2.lightlink> <45359256@news2.lightlink> <4539125e$1@news2.lightlink> <453d9a7c$1@news2.lightlink> <453e4a85@news2.lightlink> <453e824b@news2.lightlink> <453fb285@news2.lightlink> <4540f9d0@news2.lightlink> <45421333@news2.lightlink > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. >> The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. > This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: > (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. >Well, you have (8), which is kind of circular, but related. (8) simply states that if there are no balls in the vase at time t, then the vase is empty at time t; and if the there is a ball in the vase at time t, then the vase is not empty at time t. It states nothing about how that event occurs. >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. > However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. > So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. That is correct. Noon is incompatible with the problem statement. > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions Right. Your conclusion is at odds with the notion that only removals may > empty the vase, which seems to be an obvious assumption, no other means > of achieving emptiness having been mentioned. > But (T1) does /not/ merely state Only removals may empty the vase. (T1) states something quite a bit stronger: it states that if the vase becomes empty /at time t/ then removals occur /at time t/. I would formalize only removals may empty the vase (which I agree is a desirable assumption) as: (*) If, for some time t1 < t0, it is the case that the number of balls in the vase at time t1 is different than the number of balls at time t0, then there is some time t with t1 <= t <= t0, such that balls are removed at time t, or balls are added at time t. Compare (T1) and (*); they say different things. > So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? I don't believe any of those assumptions are the problem. (2) should > state that t unspoken assumption on your part which causes the problem is that noon > is part of the problem. I don't understand this complaint. Noon exists follows from (1): when we we speak of the time noon, we mean the real number 0. Do you claim that the real number 0 does not exist? And certainly noon is part of the problem: the original problem explicitly asks: What is the number of balls in the vase /at noon/? > Clearly, it cannot be, because anything that > happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem > produces a paradox by asking a question which contradicts the situation. This doesn't imply that noon doesn't occur - it simply states that the number of balls in the vase at noon cannot be determined in a well-defined manner consistent with our assumptions. But that is only the case if we assume (T1). If we /don't/ assume (T1), or we instead assume (*), then your statement does not follow; instead it follows that the vase is empty at noon. And if we /do/ accept (T1), we still have the problem I alluded to: we can /still/ prove from (1)..(8) that the problem is well-defined (empty vase at noon); but we can also prove that the problem is /not/ well-defined by the argument you give above. So something is not right with at least one of our assumptions; and the usual approach is to abandon (T1) in favor of (*). The situation here is similar to the problem: Is Socrates mortal? If we agree with /all/ of the following assumptions: (a) Socrates is a man. (b) All men are mortal. (c) No mortal lives forever. (d) Socrates lives forever through his writings. Then we can prove by (a) and (b) that Socrates is mortal; but we can also prove that Socrates is not mortal by (c) and (d). /None/ of (a)..(d) are stated /explicitly/ in the problem. A valid argument either way must be based on a /non-contradictory/ set of assumptions. So at least one of (a)..(d) must be discarded before we can claim to have made a valid argument resulting in either conclusion. Thus, if we claim by (c) and (d) that Socrates is not mortal, we are also claiming that we /don't/ agree to at least one of (a) or (b). Do you see how that relates to our discussion? === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > Well, you have (8), which is kind of circular, but related. >(8) simply states that if there are no balls in the vase at time t, > then the vase is empty at time t; and if the there is a ball in the > vase at time t, then the vase is not empty at time t. It states nothing > about how that event occurs. > What changes the number of balls, if not additions and subtractions? >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> That is correct. Noon is incompatible with the problem statement. > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> Right. Your conclusion is at odds with the notion that only removals may >> empty the vase, which seems to be an obvious assumption, no other means >> of achieving emptiness having been mentioned. >But (T1) does /not/ merely state Only removals may empty the vase. > (T1) states something quite a bit stronger: it states that if the vase > becomes empty /at time t/ then removals occur /at time t/. >I would formalize only removals may empty the vase (which I agree is > a desirable assumption) as: >(*) If, for some time t1 < t0, it is the case that the number of balls > in the vase at time t1 is different than the number of balls at time > t0, then there is some time t with t1 <= t <= t0, such that balls are > removed at time t, or balls are added at time t. And, what if, for all t in [t1,t0), there are balls? Then balls can only disappear entirely at t0. >Compare (T1) and (*); they say different things. And? Not so different after all, eh? >So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> I don't believe any of those assumptions are the problem. (2) should >> state that t> unspoken assumption on your part which causes the problem is that noon >> is part of the problem. >I don't understand this complaint. Then a dozen more explanations are a waste of time. Noon exists follows from (1): when > we we speak of the time noon, we mean the real number 0. Do you claim > that the real number 0 does not exist? And certainly noon is part of > the problem: the original problem explicitly asks: What is the number > of balls in the vase /at noon/? > Sure, the question is put in terms proscribed by the constraints of the problem. Nothing can occur at noon. It cannot become empty then, nor before. >> Clearly, it cannot be, because anything that >> happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem >> produces a paradox by asking a question which contradicts the situation. >This doesn't imply that noon doesn't occur - it simply states that > the number of balls in the vase at noon cannot be determined in a > well-defined manner consistent with our assumptions. Then you admit you have no answer, given your formulation? >But that is only the case if we assume (T1). If we /don't/ assume (T1), > or we instead assume (*), then your statement does not follow; instead > it follows that the vase is empty at noon. So, you don't want to assume that the only way balls can leave the vase is by removal, or that the vase can only become empty by balls leaving it? Which of those would you like to reject? >And if we /do/ accept (T1), we still have the problem I alluded to: we > can /still/ prove from (1)..(8) that the problem is well-defined (empty > vase at noon); but we can also prove that the problem is /not/ > well-defined by the argument you give above. That's what I was saying. Oy. Noon is not consistent with the problem. The question is nonsensical given the parameters. >So something is not right with at least one of our assumptions; and the > usual approach is to abandon (T1) in favor of (*). Yeah. Noon doesn't happen. >The situation here is similar to the problem: >Is Socrates mortal? O God, here we go... >If we agree with /all/ of the following assumptions: >(a) Socrates is a man. > (b) All men are mortal. > (c) No mortal lives forever. > (d) Socrates lives forever through his writings. >Then we can prove by (a) and (b) that Socrates is mortal; but we can > also prove that Socrates is not mortal by (c) and (d). Depends on c's qualifications. >/None/ of (a)..(d) are stated /explicitly/ in the problem. A valid > argument either way must be based on a /non-contradictory/ set of > assumptions. So at least one of (a)..(d) must be discarded before we > can claim to have made a valid argument resulting in either conclusion. >Thus, if we claim by (c) and (d) that Socrates is not mortal, we are > also claiming that we /don't/ agree to at least one of (a) or (b). >Do you see how that relates to our discussion? > Very slightly... === Subject: Re: An uncountable countable set > What changes the number of balls, if not additions and subtractions? Insertions and removals! > And, what if, for all t in [t1,t0), there are balls? Then balls can only > disappear entirely at t0. Does TO object to having a real function, f, such that f(x) = 0 if and only if x >= 0? > Sure, the question is put in terms proscribed by the constraints of the > problem. Nothing can occur at noon. It cannot become empty then, nor before. That is like saying f(x) = x cannot be 0 at x = 0 because it is not zero anywhere else. > That's what I was saying. Oy. Noon is not consistent with the problem. > The question is nonsensical given the parameters. The only parameter is time, and every time is measured from what TO claims does not exist. >Yeah. Noon doesn't happen. Then no part of the problem can happen . The only relevant question is According to the rules set up in the problem, is each ball which is inserted into the vase before noon also removed from the vase before noon? An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? === Subject: Re: An uncountable countable set <45341984@news2.lightlink> <45359256@news2.lightlink> <4539125e$1@news2.lightlink> <453d9a7c$1@news2.lightlink> <453e4a85@news2.lightlink> <453e824b@news2.lightlink> <453fb285@news2.lightlink> <4540f9d0@news2.lightlink> <45421333@news2.lightlink> Forgive me if I blunder in on Chas's carefully constructed argument, but... > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. >> The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. > This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: > (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. >Well, you have (8), which is kind of circular, but related. >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. > However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. > So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. That is correct. Noon is incompatible with the problem statement. > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions Right. Your conclusion is at odds with the notion that only removals may > empty the vase, which seems to be an obvious assumption, no other means > of achieving emptiness having been mentioned. > So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? I don't believe any of those assumptions are the problem. (2) should > state that t unspoken assumption on your part which causes the problem is that noon > is part of the problem. Clearly, it cannot be, because anything that > happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem > produces a paradox by asking a question which contradicts the situation. > Nothing happens at noon. The process never completes the unending set. Here's something I don't understand. I believe, Tony, that you think that if every one of these pofnat-labelled balls is inserted one minute earlier (so *informally*, instead of a sliver tapering to zero width, we have an endless boomerang shape, with the width tending to 1 as you go ever up the y-direction), then at noon no balls are left. Presumably because once all the balls are IN (at 11:59), there is only removal, tick, tick, tick, ... and all are gone at noon. But why doesn't this stuff about noon being incompatible apply here too? Is there a *principled* way in which you determine which arguments apply at particular points? (I'm sure it appears to most non-cranks here that there isn't.) Note that in this scenario, at time noon- 1/n, there are, da-dah!, an infinite number of balls in the vase. So the limit of the number of balls in the vase at t approaches noon is infinity. Yet you (really?) think that in this case the vase ends up empty? Do you have any sort of *mathematical* argument for this (as opposed to intuition and hand-waving?) Brian Chandler http://imaginatorium.org === Subject: Re: An uncountable countable set > Forgive me if I blunder in on Chas's carefully constructed argument, > but... > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > Well, you have (8), which is kind of circular, but related. >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> That is correct. Noon is incompatible with the problem statement. > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> Right. Your conclusion is at odds with the notion that only removals may >> empty the vase, which seems to be an obvious assumption, no other means >> of achieving emptiness having been mentioned. > So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> I don't believe any of those assumptions are the problem. (2) should >> state that t> unspoken assumption on your part which causes the problem is that noon >> is part of the problem. Clearly, it cannot be, because anything that >> happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem >> produces a paradox by asking a question which contradicts the situation. >> Nothing happens at noon. The process never completes the unending set. >Here's something I don't understand. I believe, Tony, that you think > that if every one of these pofnat-labelled balls is inserted one minute > earlier (so *informally*, instead of a sliver tapering to zero width, > we have an endless boomerang shape, with the width tending to 1 as you > go ever up the y-direction), then at noon no balls are left. Presumably > because once all the balls are IN (at 11:59), there is only removal, > tick, tick, tick, ... and all are gone at noon. But why doesn't this > stuff about noon being incompatible apply here too? Is there a > *principled* way in which you determine which arguments apply at > particular points? (I'm sure it appears to most non-cranks here that > there isn't.) That's very simple, Brian. The limit of balls as n->noon is 0. That's not the case in the original problem. There, there is no limit. The sum diverges, as it does in this case until 11:59. Those points of infinitely quick iterations ultimately include an uncountable number of iterations, unless their countability is specified, in which case they do not reach those points of uncountability. Additionally, we have the fact that, if property p applies at all times before time t, and does not change state at time t, then it continues to apply at time t. In the case where we have a countably infinite number of balls at t=-1, no matter how they got there, and start removing them in Zeno fashion, we can conceptually empty the vase by time 0. Once time 0 is there, nothing else happens. So if all balls have been removed by then, that's the way it is at time 0. If all balls haven't been removed, due to a condition of the problem under consideration, then it's not empty. All balls have NOT been removed before noon in the gedanken. AT noon, no balls are removed. The vase can only be empty after having been non-empty if removals have occurred between those two times. Which of my statements above do you find objectionable, and why? That would be helpful to know. Tony >Note that in this scenario, at time noon- 1/n, there are, da-dah!, an > infinite number of balls in the vase. So the limit of the number of > balls in the vase at t approaches noon is infinity. Yet you (really?) > think that in this case the vase ends up empty? Do you have any sort of > *mathematical* argument for this (as opposed to intuition and > hand-waving?) >Brian Chandler > http://imaginatorium.org > oo=oo -> lim(x->oo: oo-x)=0. If your infinities are consistent, then you have removed every element in the set. What makes you think lim(x->oo: oo-x)=oo, when x-x=0 and x=oo??? I know you think my thinking here is simplistic. It's simply not unnecessarily complicated and confused. === Subject: Re: An uncountable countable set > Forgive me if I blunder in on Chas's carefully constructed argument, > but... >Here's something I don't understand. I believe, Tony, that you think > that if every one of these pofnat-labelled balls is inserted one minute > earlier (so *informally*, instead of a sliver tapering to zero width, > we have an endless boomerang shape, with the width tending to 1 as you > go ever up the y-direction), then at noon no balls are left. Presumably > because once all the balls are IN (at 11:59), there is only removal, > tick, tick, tick, ... and all are gone at noon. But why doesn't this > stuff about noon being incompatible apply here too? Is there a > *principled* way in which you determine which arguments apply at > particular points? (I'm sure it appears to most non-cranks here that > there isn't.) >That's very simple, Brian. The limit of balls as n->noon is 0. That's > not the case in the original problem. So that TO is arguing that the later balls are put in, the more will be there at noon? But that also creates contradictions. Let us delay the insertion of ball number n, which is to be removed at 1/n minutes before noon until (1/n + 1/(n-1))/2 minutes before noon, but remove it as originally scheduled. Then at any time before noon there will be either 0 or 1 balls in the vase, and these values will alternate. If TO's logic were consistent, then at noon there would have to be half a ball in the vase, but what number, or fraction of a number, it would bear, or what numbers it would share, is not readily apparent. >Which of my statements above do you find objectionable, and why? That > would be helpful to know. Among others, the notion that balls are created at noon out of thin air to fill the vacuum that having every ball removed before noon seems to create in TO's head. === Subject: Re: An uncountable countable set > Here's something I don't understand. I believe, Tony, that you think > that if every one of these pofnat-labelled balls is inserted one minute > earlier (so *informally*, instead of a sliver tapering to zero width, > we have an endless boomerang shape, with the width tending to 1 as you > go ever up the y-direction), then at noon no balls are left. Are you saying that for n = 1,2,..., we should let A_n = -1/floor((n+9)/10) - 1, R_n = -1/n ? > Presumably > because once all the balls are IN (at 11:59), there is only removal, > tick, tick, tick, ... and all are gone at noon. But why doesn't this > stuff about noon being incompatible apply here too? Is there a > *principled* way in which you determine which arguments apply at > particular points? (I'm sure it appears to most non-cranks here that > there isn't.) >Note that in this scenario, at time noon- 1/n, there are, da-dah!, an > infinite number of balls in the vase. So the limit of the number of > balls in the vase at t approaches noon is infinity. Yet you (really?) > think that in this case the vase ends up empty? Do you have any sort of > *mathematical* argument for this (as opposed to intuition and > hand-waving?) -- David Marcus === Subject: Re: An uncountable countable set This is very simple. Everything that occurs is either an addition of ten > balls or a removal of 1, and occurs a finite amount of time before noon. > At the time of each event, balls remain. At noon, no balls are inserted or > removed. The vase can only become empty through the removal of balls, so > if no balls are removed, the vase cannot become empty at noon. It was not > empty before noon, therefore it is not empty at noon. Nothing can happen > at noon, since that would involve a ball n such that 1/n=0. Tony, I think your confusion results from imagining the balls without any labels. In this case at 1 minute before noon 10 balls are inserted into the vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 minute before noon, 9 more balls are inserted into the vase and, in general, at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you are saying that the number of vase balls at noon is: 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. Or, since one ball is removed each time ten more are added, we should write: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. Now, this divergent series is conditionally convergent. That means we can make the sum equal any value we like depending on how the terms are arranged. So if we choose 0 for the sum that is perfectly valid: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. In this case there are no balls in the vase at noon. Without labels on the balls there is no criterion by which to select what the sum should be and the end state of the supertask is undefined. As I noted in an earlier post, if some of the balls are labeled with numbers that are not naturals, for example transfinite ordinal numbers, we can choose Infinite for the sum if the circumstances require it. Consider the following problem: Tony has a two gallon bucket and his job is to ensure that the amount of water in the bucket during the nth day is 1+sin(n) gallons. Since Tony's job never ends he will always be making daily changes in the bucket's water content and we have a full mathematical description of Tony's job. There is no problem with this. But if we changed Tony's job so that it had an end, say at noon, and the bucket had to contain 1+sin(n) gallons at (1/2)^n minutes before noon then we do not have a full description of Tony's activities. It is a mistake to assume the bucket's water content at noon is a function of its pre-noon state. At noon Tony puts whatever amount of water he wants into the bucket. -R === Subject: Re: An uncountable countable set > This is very simple. Everything that occurs is either an addition of ten > balls or a removal of 1, and occurs a finite amount of time before noon. > At the time of each event, balls remain. At noon, no balls are inserted or > removed. The vase can only become empty through the removal of balls, so > if no balls are removed, the vase cannot become empty at noon. It was not > empty before noon, therefore it is not empty at noon. Nothing can happen > at noon, since that would involve a ball n such that 1/n=0. Tony, I think your confusion results from imagining the balls without any > labels. In this case at 1 minute before noon 10 balls are inserted into the > vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 > minute before noon, 9 more balls are inserted into the vase and, in general, > at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you > are saying that the number of vase balls at noon is: 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. Or, since one ball is removed each time ten more are added, we should write: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. Now, this divergent series is conditionally convergent. That means we can > make the sum equal any value we like depending on how the terms are > arranged. So if we choose 0 for the sum that is perfectly valid: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. In this case there are no balls in the vase at noon. Without labels on the > balls there is no criterion by which to select what the sum should be and > the end state of the supertask is undefined. As I noted in an earlier post, > if some of the balls are labeled with numbers that are not naturals, for > example transfinite ordinal numbers, we can choose Infinite for the sum if > the circumstances require it. Consider the following problem: Tony has a two gallon bucket and his job is to ensure that the amount of > water in the bucket during the nth day is 1+sin(n) gallons. Since Tony's > job never ends he will always be making daily changes in the bucket's water > content and we have a full mathematical description of Tony's job. There is > no problem with this. But if we changed Tony's job so that it had an end, > say at noon, and the bucket had to contain 1+sin(n) gallons at (1/2)^n > minutes before noon then we do not have a full description of Tony's > activities. It is a mistake to assume the bucket's water content at noon is > a function of its pre-noon state. At noon Tony puts whatever amount of > water he wants into the bucket. Although one might make a reasonable argument that the value should b e between 0 and 2 gallons, inclusive. === Subject: Re: An uncountable countable set >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted or >> removed. The vase can only become empty through the removal of balls, so >> if no balls are removed, the vase cannot become empty at noon. It was not >> empty before noon, therefore it is not empty at noon. Nothing can happen >> at noon, since that would involve a ball n such that 1/n=0. Tony, I think your confusion results from imagining the balls without any > labels. In this case at 1 minute before noon 10 balls are inserted into the > vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 > minute before noon, 9 more balls are inserted into the vase and, in general, > at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you > are saying that the number of vase balls at noon is: 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. > Yes, but I don't consider that confusion. If the problem is solvable without the labels, then the labels don't matter. >Or, since one ball is removed each time ten more are added, we should write: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. Now, this divergent series is conditionally convergent. That means we can > make the sum equal any value we like depending on how the terms are > arranged. So if we choose 0 for the sum that is perfectly valid: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. > No, we went through this in another thread. The only way to get a sum of 0 is by rearranging the terms and grouping so you have ten -1's for every +10. But, the sequence of events is specified NOT to be in that order. No ball can be removed without having ten inserted immediately before. So, despite the silly games that mathematicians may play with conditionally convergent series, none of that applies to the ball and vase problem as stated. Does that sound confused to you? >In this case there are no balls in the vase at noon. Without labels on the > balls there is no criterion by which to select what the sum should be and > the end state of the supertask is undefined. As I noted in an earlier post, > if some of the balls are labeled with numbers that are not naturals, for > example transfinite ordinal numbers, we can choose Infinite for the sum if > the circumstances require it. > No, the order cannot be rearranged. For each iteration you have a net addition of nine balls. You cannot remove a ball without adding ten more. This is clearly the divergent sum(n=1->oo: 9). >Consider the following problem: Tony has a two gallon bucket and his job is to ensure that the amount of > water in the bucket during the nth day is 1+sin(n) gallons. Since Tony's > job never ends he will always be making daily changes in the bucket's water > content and we have a full mathematical description of Tony's job. There is > no problem with this. But if we changed Tony's job so that it had an end, > say at noon, and the bucket had to contain 1+sin(n) gallons at (1/2)^n > minutes before noon then we do not have a full description of Tony's > activities. It is a mistake to assume the bucket's water content at noon is > a function of its pre-noon state. At noon Tony puts whatever amount of > water he wants into the bucket. -R Now you are talking about a series which diverges due to oscillation, more or less. At noon the bucket would have to be filling and emptying infinitely quickly. So what? Clearly, at noon, it has somewhere between 0 and 2 gallons of water, but no specific quantity. That's a different problem. === Subject: Re: An uncountable countable set >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted or >> removed. The vase can only become empty through the removal of balls, so >> if no balls are removed, the vase cannot become empty at noon. It was not >> empty before noon, therefore it is not empty at noon. Nothing can happen >> at noon, since that would involve a ball n such that 1/n=0. Tony, I think your confusion results from imagining the balls without any > labels. In this case at 1 minute before noon 10 balls are inserted into > the > vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 > minute before noon, 9 more balls are inserted into the vase and, in > general, > at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you > are saying that the number of vase balls at noon is: 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. Yes, but I don't consider that confusion. If the problem is solvable > without the labels, then the labels don't matter. But the problem is NOT solvable without labels if that means being able to determine what is left in the vase at noon. Or, since one ball is removed each time ten more are added, we should > write: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. Now, this divergent series is conditionally convergent. That means we can > make the sum equal any value we like depending on how the terms are > arranged. So if we choose 0 for the sum that is perfectly valid: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. No, we went through this in another thread. The only way to get a sum of > 0 is by rearranging the terms and grouping so you have ten -1's for > every +10. But, the sequence of events is specified NOT to be in that > order. No ball can be removed without having ten inserted immediately > before. And no ball can be inserted without being removed before noon. > So, despite the silly games that mathematicians may play with > conditionally convergent series, none of that applies to the ball and > vase problem as stated. Does that sound confused to you? No more so that TO's usual level of total confusion. In this case there are no balls in the vase at noon. Without labels on the > balls there is no criterion by which to select what the sum should be and > the end state of the supertask is undefined. As I noted in an earlier > post, > if some of the balls are labeled with numbers that are not naturals, for > example transfinite ordinal numbers, we can choose Infinite for the sum > if > the circumstances require it. . You cannot remove a ball without adding ten > more. And you also cannot insert a ball without removing it before noon. === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. >No one disagrees with the above statements. > >> The vase can only become empty through the removal of balls, >Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >(T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > >> so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. >Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? > and I'm rather tired right now, but at first glance it seems like it could be a sound analysis. I've cut and pasted for perusal when I'm sharper tomorrow. === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. >Here's some of my thoughts: >When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t That doesn't mean t has to be able to assume ALL real numbers. The times in [-1,0) are all real numbers. >When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. No, I accept that. There is no time after t=-1 where there is no ball in the vase. >When you say an infinite number of balls are removed at time 0, I > think he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff). I didn't say that exactly. If 0 occurs, then all finite balls are gone, but infinite balls have been inserted such that 1/n=0 for those balls. So, at noon the vase is not empty, even if it occurs in the problem, which it doesn't. >All these assertions follow a simgle theme: If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?. > That there is a contradiction in your conclusion if you assume that all events must occur at some time, and that becoming empty is the result of events that happen in the vase. It cannot become empty until noon, when nothing happens to cause it. === Subject: Re: An uncountable countable set <45359256@news2.lightlink> <4539125e$1@news2.lightlink> <453d9a7c$1@news2.lightlink> <453e4a85@news2.lightlink> <453e824b@news2.lightlink> <453fb285@news2.lightlink> <4540f9d0@news2.lightlink> <4541825d@news2.lightlink> <4542201a@news2.lightlink > > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. > Here's some of my thoughts: > When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t That doesn't mean t has to be able to assume ALL real numbers. The times > in [-1,0) are all real numbers. And I would say that assuming that by a time t, we mean a real number in [-1,0) is a different assumption than (1). > When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. No, I accept that. There is no time after t=-1 where there is no ball in > the vase. > I.e., there is a ball in the vase. But then by the argument I previously gave, there is then a ball in the vase which is not in the vase. Your reference to unspecified balls in the vase at noon I interpret to be a way of saying that (8) should instead state something like if the number of balls in the vase is not 0, then there may be no /specific/ ball in the vase (because there is instead an /unspecific/ ball in the vase). > When you say an infinite number of balls are removed at time 0, I > think he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff). I didn't say that exactly. If 0 occurs, then all finite balls are gone, > but infinite balls have been inserted such that 1/n=0 for those balls. > So, at noon the vase is not empty, even if it occurs in the problem, > which it doesn't. > If infinite balls are inserted at some time t = -1/n = 0, then by (5) each of them are inserted at time t; and at that time exactly 10 balls are inserted. 10 is not infinite. > All these assertions follow a simgle theme: If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?. > > That there is a contradiction in your conclusion if you assume that all > events must occur at some time... The occurence of these events (ball insertions and removals at particular times) is described by (1), (5), (6), and (7). > ... and that becoming empty is the result of > events that happen in the vase. There is no becoming empty described in (1)..(8). There is only being empty; which is described by (1), (2), (3), and (4), and (8). > It cannot become empty until noon, when > nothing happens to cause it. And that is your premise that I call (T1). It is incompatible with (1)..(8); so either we must reject something in (1)..(8), or we must reject (T1) in favor of my previously described (*). === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. > Here's some of my thoughts: >> When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t >> That doesn't mean t has to be able to assume ALL real numbers. The times >> in [-1,0) are all real numbers. >And I would say that assuming that by a time t, we mean a real number > in [-1,0) is a different assumption than (1). > There is no contradiction between them, is there.... > When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. >> No, I accept that. There is no time after t=-1 where there is no ball in >> the vase. >I.e., there is a ball in the vase. But then by the argument I > previously gave, there is then a ball in the vase which is not in the > vase. Your reference to unspecified balls in the vase at noon I > interpret to be a way of saying that (8) should instead state something > like if the number of balls in the vase is not 0, then there may be no > /specific/ ball in the vase (because there is instead an /unspecific/ > ball in the vase). > You claim no balls are added at noon, because nothing can be, but then, nothing can be removed at noon, either. Either it grows or stays un-zero. > When you say an infinite number of balls are removed at time 0, I > think he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff). >> I didn't say that exactly. If 0 occurs, then all finite balls are gone, >> but infinite balls have been inserted such that 1/n=0 for those balls. >> So, at noon the vase is not empty, even if it occurs in the problem, >> which it doesn't. >If infinite balls are inserted at some time t = -1/n = 0, then by (5) > each of them are inserted at time t; and at that time exactly 10 balls > are inserted. 10 is not infinite. > It i larger than 1, and at an infinite rate of insertion, yes, an infinite number of balls are added at t=0. > All these assertions follow a simgle theme: If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?. >> That there is a contradiction in your conclusion if you assume that all >> events must occur at some time... >The occurence of these events (ball insertions and removals at > particular times) is described by (1), (5), (6), and (7). > >> ... and that becoming empty is the result of >> events that happen in the vase. >There is no becoming empty described in (1)..(8). There is only > being empty; which is described by (1), (2), (3), and (4), and (8). > Put it in there, in whatever form you think might be correct. I assure you, it will make you question your conclusions. Then, there is a question of logical validity there. We have to ascertain to root of the problem. Or, we don't. We can just stay dumb. >> It cannot become empty until noon, when >> nothing happens to cause it. >And that is your premise that I call (T1). It is incompatible with > (1)..(8); so either we must reject something in (1)..(8), or we must > reject (T1) in favor of my previously described (*). > What exactly is the contradiction, specifically? Can you formulate that? === Subject: Re: An uncountable countable set > You claim no balls are added at noon, because nothing can be, but then, > nothing can be removed at noon, either. Either it grows or stays un-zero. How un-zero when every ball is removed before noon? The only relevant question is According to the rules set up in the problem, is each ball which is inserted into the vase before noon also removed from the vase before noon? An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. > yes, an > infinite number of balls are added at t=0. Where do these infinitely many balls come from? As all the numbered balls have been removed by noon and there is no provision in the gedankenexperiment for the existence, much less the insertion, of any others, from what twilight zone does TO produce his phantom unnumbered balls? Whatever TO is smoking must be powerful stuff to create something out of nothing. > We can just stay dumb. TO certainly can. > What exactly is the contradiction, specifically? Can you formulate that? There is no contradiction involved in the vase being empty at noon, but many for it not being empty at noon. === Subject: Re: An uncountable countable set > When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t >That doesn't mean t has to be able to assume ALL real numbers. The times > in [-1,0) are all real numbers. By what mechanism does TO propose to stop time? When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. >No, I accept that. There is no time after t=-1 where there is no ball in > the vase. That is not the same thing at all, as it requires that some ball remain in the vase after it has been removed. > I didn't say that exactly. If 0 occurs, then all finite balls are gone, As those are the only balls that the gedankenexperiment allows, that means the vase is then empty. > but infinite balls have been inserted Where in the original gedankenexperiment is there any provision made for those alleged infinite balls? What TO does is decide what result he wants and then tries to bend the facts to fit. But it does not work. The only relevant question is According to the rules of the gedankenexperiment , is each ball which is inserted into the vase before noon also removed from the vase before noon? An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the gedankenexperiment. So TO keeps violating the conditions of the gedankenexperiment. === Subject: Re: An uncountable countable set When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t >> That doesn't mean t has to be able to assume ALL real numbers. The times >> in [-1,0) are all real numbers. >By what mechanism does TO propose to stop time? By the mechanism of unfinishablility. > When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. >> No, I accept that. There is no time after t=-1 where there is no ball in >> the vase. >That is not the same thing at all, as it requires that some ball remain > in the vase after it has been removed. No it requires that some ball e=remain after some other ball is removed. > >> I didn't say that exactly. If 0 occurs, then all finite balls are gone, >As those are the only balls that the gedankenexperiment allows, that > means the vase is then empty. > And so, nothing can happen at noon. >> but infinite balls have been inserted >Where in the original gedankenexperiment is there any provision made for > those alleged infinite balls? At noon. >What TO does is decide what result he wants and then tries to bend the > facts to fit. Does anything happen at noon? >But it does not work. Does the vase empty before, or at, noon? >The only relevant question is According to the rules of the > gedankenexperiment , is each ball which is inserted into the vase before > noon also removed from the vase before noon? >An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the > gedankenexperiment. >So TO keeps violating the conditions of the gedankenexperiment. Nope. I keep obeying them. === Subject: Re: An uncountable countable set When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t >> That doesn't mean t has to be able to assume ALL real numbers. The times >> in [-1,0) are all real numbers. >By what mechanism does TO propose to stop time? >By the mechanism of unfinishablility. That TO cannot finish something, does not mean that it is unfinishable. It just means that TO is incompetent, which we already knew. >When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. >> No, I accept that. There is no time after t=-1 where there is no ball in >> the vase. >That is not the same thing at all, as it requires that some ball remain > in the vase after it has been removed. >No it requires that some ball e=remain after some other ball is removed. That requires that some ball remains when all balls, including it, have been removed. > I didn't say that exactly. If 0 occurs, then all finite balls are gone, >As those are the only balls that the gedankenexperiment allows, that > means the vase is then empty. And so, nothing can happen at noon. > >> but infinite balls have been inserted >Where in the original gedankenexperiment is there any provision made for > those alleged infinite balls? >At noon. Where does it say so? What TO does is decide what result he wants and then tries to bend the > facts to fit. >Does anything happen at noon? The vase becomes empty. But it does not work. >Does the vase empty before, or at, noon? Since it is empty AT noon, becoming empty is of no importance. The only relevant question is According to the rules of the > gedankenexperiment , is each ball which is inserted into the vase before > noon also removed from the vase before noon? >An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the > gedankenexperiment. >So TO keeps violating the conditions of the gedankenexperiment. >Nope. I keep obeying them. TO is obeying rules that he assumes out of nowhere but which are no part of the original statement of the gedankenexperiment. In the original, each ball inserted before noon is removed at a time before noon. So where do the mythical balls that TO claims are in the vase at noon to come from? No source for them is justifiable from any part of the original gedankenexperiment. === Subject: Re: An uncountable countable set <45359256@news2.lightlink> <4539125e$1@news2.lightlink> <453d9a7c$1@news2.lightlink> <453e4a85@news2.lightlink> <453e824b@news2.lightlink> <453fb285@news2.lightlink> <4540f9d0@news2.lightlink> <4541825d@news2.lightlink> <4542201a@news2.lightlink > > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. > Here's some of my thoughts: > When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t That doesn't mean t has to be able to assume ALL real numbers. The times > in [-1,0) are all real numbers. And I would say that assuming that by a time t, we mean a real number in [-1,0) is a different assumption than (1). > When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. No, I accept that. There is no time after t=-1 where there is no ball in > the vase. > I.e., there is a ball in the vase. But then by the argument I previously gave, there is then a ball in the vase which is not in the vase. Your reference to unspecified balls in the vase at noon I interpret to be a way of saying that (8) should instead state something like if the number of balls in the vase is not 0, then there may be no /specific/ ball in the vase (because there is instead an /unspecific/ ball in the vase). > When you say an infinite number of balls are removed at time 0, I > think he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff). I didn't say that exactly. If 0 occurs, then all finite balls are gone, > but infinite balls have been inserted such that 1/n=0 for those balls. > So, at noon the vase is not empty, even if it occurs in the problem, > which it doesn't. > If infinite balls are inserted at some time t = -1/n = 0, then by (5) each of them are inserted at time t; and at that time exactly 10 balls are inserted. 10 is not infinite. > All these assertions follow a simgle theme: If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?. > > That there is a contradiction in your conclusion if you assume that all > events must occur at some time... The occurence of these events (ball insertions and removals at particular times) is described by (1), (5), (6), and (7). > ... and that becoming empty is the result of > events that happen in the vase. There is no becoming empty described in (1)..(8). There is only being empty; which is described by (1), (2), (3), and (4), and (8). > It cannot become empty until noon, when > nothing happens to cause it. And that is your premise that I call (T1). It is incompatible with (1)..(8); so either we must reject something in (1)..(8), or we must reject (T1) in favor of my previously described (*). === Subject: Re: An uncountable countable set > My question : what do you think is in the vase at noon? > A countable infinity of balls. >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted >> or removed. > No one disagrees with the above statements. > The vase can only become empty through the removal of balls, > Note that this is not identical to saying the vase can only become > empty /at time t/, if there are balls removed /at time t/; which is > what it seems you actually mean. >> This doesn't follow from (1)..(8), which lack any explicit mention of > what becomes empty means. However, we can easily make it an > assumption: >> (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > so if no balls are removed, the vase cannot become empty at noon. It was >> not empty before noon, therefore it is not empty at noon. Nothing can >> happen at noon, since that would involve a ball n such that 1/n=0. > Now your logical argument is complete, assuming we also accept > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > (6), the number of balls changes at time 0; and therefore by (T1), > balls are either placed or removed at time 0, implying by (5) and (6) > that there is a natural number n such that -1/n = 0; which is absurd. > Therefore, by reductio ad absurdum, the number of balls at time 0 > cannot be 0. >> However, it does not follow that the number of balls in the vase is > therefore any other natural number n, or even infinite, at time 0; > because that would /equally/ require that the number of balls changes > at time 0, and that in turn requires by (T1) that balls are either > added or removed at time 0; and again by (5) or (6) this implies that > there is a natural number n with -1/n = 0; which is absurd. So again, > we get that any statement of the form the number of balls at time 0 is > (anything) must be false by reductio absurdum. >> So if we include (T1) as an assumption as well as (1)..(8), it follows > logically that the number of balls in the vase at time 0 is not > well-defined. >> Of course, we also find that by (1)..(8) and (T1), it /still/ follows > logically that the number of balls in the vase at time t is 0; and this > is a problem: we can prove two different and incompatible statements > from the same set of assumptions >> So at least one of the assumptions (1)..(8) and (T1) must be discarded > if we are to resolve this. What do you suggest? Which of (1)..(8) do > you want discard to maintain (T1)? >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. > Here's some of my thoughts: >> When you say noon doesn't occur; I think he doesn't accept (1): by a > time t, we mean a real number t >> That doesn't mean t has to be able to assume ALL real numbers. The times >> in [-1,0) are all real numbers. >And I would say that assuming that by a time t, we mean a real number > in [-1,0) is a different assumption than (1). >When you say if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0, I think he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase. >> No, I accept that. There is no time after t=-1 where there is no ball in >> the vase. >I.e., there is a ball in the vase. But then by the argument I > previously gave, there is then a ball in the vase which is not in the > vase. Your reference to unspecified balls in the vase at noon I > interpret to be a way of saying that (8) should instead state something > like if the number of balls in the vase is not 0, then there may be no > /specific/ ball in the vase (because there is instead an /unspecific/ > ball in the vase). > Specify the largest natural, and I'll grant you that. > When you say an infinite number of balls are removed at time 0, I > think he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff). >> I didn't say that exactly. If 0 occurs, then all finite balls are gone, >> but infinite balls have been inserted such that 1/n=0 for those balls. >> So, at noon the vase is not empty, even if it occurs in the problem, >> which it doesn't. >If infinite balls are inserted at some time t = -1/n = 0, then by (5) > each of them are inserted at time t; and at that time exactly 10 balls > are inserted. 10 is not infinite. 10 balls per iteration times oo iterations per second is oo (or 10*oo) balls per second. >All these assertions follow a simgle theme: If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?. >> That there is a contradiction in your conclusion if you assume that all >> events must occur at some time... >The occurence of these events (ball insertions and removals at > particular times) is described by (1), (5), (6), and (7). There is the event of becoming empty, i.e., in(t)-out(t)=0. Oh, except, that never happens. > >> ... and that becoming empty is the result of >> events that happen in the vase. >There is no becoming empty described in (1)..(8). There is only > being empty; which is described by (1), (2), (3), and (4), and (8). > Formulate it, and behold your error. >> It cannot become empty until noon, when >> nothing happens to cause it. >And that is your premise that I call (T1). It is incompatible with > (1)..(8); so either we must reject something in (1)..(8), or we must > reject (T1) in favor of my previously described (*). > Why exactly is it inconsistent with (1)-(8)? Which premise does it contradict? Please specify. === Subject: Re: An uncountable countable set > Specify the largest natural, and I'll grant you that. TO on his largest natural kick again. When TO can produce 4 sided triangles on request, or points of intersection of parallel lines in Euclidean spaces, only then will he have the right to ask for largest naturals. > The occurence of these events (ball insertions and removals at > particular times) is described by (1), (5), (6), and (7). >There is the event of becoming empty, i.e., in(t)-out(t)=0. Oh, except, > that never happens. Except that if it doesn't happen nothing else can happen either, since without noon, there is no forenoon. >There is no becoming empty described in (1)..(8). There is only > being empty; which is described by (1), (2), (3), and (4), and (8). Formulate it, and behold your error. We formulated it and beheld a lot of TO's errors but none of our own. The only relevant question is According to the rules set up in the gedankenexperiment, is each ball which is inserted into the initially empty vase before noon also removed from the vase before noon? An affirmative answer guarantees that the vase is empty at noon. A negative answer directly violates the conditions of the problem. So, of course, TO answers negatively. === Subject: Re: An uncountable countable set >> says. But Tony, I have a question for you. Suppose we put one more ball > into the vase, at any time before noon, and that ball is labeled oo. > At exactly noon that ball is removed from the vase. At noon is the vase > empty or does it contain the ball labeled oo? >> -R > Hi RLG. Welcome to the conversation. >> According to the experiment, all balls inserted and removed are finite, so >> that doesn't really apply. Every ball n is inserted at time -1/n >> (or -1/2^n originally, but it doesn't matter), so ball oo cannot be >> inserted before noon. >> But, if you want to entertain the idea of inserting an extra ball named >> oo, or Bill, or RLG, the addition of a ball is not going to make the >> vase any more empty. I wonder what logical implication you think that >> has.... >My apologies if the logical implication was unclear. What I was trying to > get at was the issue of how well point set theory does or does not apply to > the continuum. Suppose a ball labeled oo is put into the vase five > minutes before noon. I asked that if at exactly noon the ball labeled oo > is removed from the vase is the vase empty at noon or does it contain the > ball labeled oo? Is the vase time for the oo ball [-5,0] or [-5,0) and > is there a difference between these two in terms of temporal duration? In > the [-5,0] case the ball is in the vase at the instant t=0 but not at any > time t>0. Since the ball is in the vase at t=0 it is not removed at that > time so it must be removed at some time t>0. But for every t>0 there is a > (t/2)>0 such that t>(t/2)>0 and so at no t>0 is the ball removed from the > vase. It seems that in the [-5,0] case there is no instant at which the > oo ball is removed from the vase. In the [-5,0) case the ball is not in > the vase at time t=0 so it seems it should have been removed at some time > t<0. But the oo ball is in the vase for all t<0 and so at no t<0 is the > oo ball removed from the vase. It seems that in the [-5,0) case there is > no instant at which the oo ball is removed from the vase. So I ask you > again, if the oo ball is removed from the vase at exactly noon, is it in > the vase at exactly noon or not? Maybe this is a joke question. >Before considering that question too deeply, consider another question. The > length of the two intervals [0,1] and [0,1) is both 1. In point set theory > these are two different sets, the latter set is identical to the former set > except it does not contain the number 1. Yet, in terms of distance, these > distinctions make no sense. If you have a stick one meter long and take > nothing at all off its right end, you haven't changed anything. In terms of > distance and time distinctions like [-5,0], [-5,0), [0,1], and [0,1) are > meaningless. This raises the question as to how well point set theory > applies to the continuum and I remember reading somewhere that Godel had > concerns about this. In my opinion, point set theory is the best tool > presently available for studying the continuum but I recognize that there > are some limitations to it. I share your and Godel's concerns about point set theory, or at least have some concerns of my own, which arose in a conversation regarding the diagonal from (0,0) to (1,1) as the limit of a staircase between those two points, as the number of steps approached oo, since the points between the two objects become arbitrarily close. This was Chas' counterexample to my claim that inductive proof can be extended to the infinite case, with certain precautions. His claim was that, if it were true, then the diagonal should have length 2, as do all the finite staircases. My counterclaim was that the staircase in the limit was not the same object as the diagonal, but was a fractal line, maintaining its right angle on an infinitesimal scale. To demonstrate, I concocted a segment sequence topology which clearly showed the difference between the two. So, yes, I agree there are issues with point set approaches when it comes to measure. You mention measure above, in terms of [0,1] and [0,1) having the same measure, but being different sets. I actually consider the second to have infinitesimally smaller measure than the first, a difference of the unit infinitesimal. Applying this notion to the experiment under discussion, I would say the experiment occurs in [-1,0) and that 0 is not included, or else n must become infinite, which is not allowed. If one is to entertain the notion of noon in the experiment, and claim the vase is empty at noon, it had to have become empty, either before noon, or at noon. When it comes to the question of whether a ball is in or out at its moment of removal, it's both for that instant. We can agree to a convention that addition or removal affects the vase from that moment forward, or for all subsequent moments. I don't think it makes a difference. The event still occurred at that moment, right? >This whole issue that you and other have been arguing is called the > Ross-Littlewood paradox. I suggest you read about it at > http://en.wikipedia.org/wiki/Supertask. Wikipedia phrases it this way: >Suppose you had a jar capable of containing infinitely many marbles, and an > infinite collection of marbles labeled 1, 2, 3, and so on. At t=0, marbles 1 > to 10 are placed in the jar, at t=1/2 11 to 20 are placed in the jar but > marble 1 is taken out. At t=3/4 marbles 21 to 30 are put in the jar and > marble 2 is taken out: in general at time t=1-(1/2)^n, the marbles (10*n + > 1) to (10*n + 10) are placed in the jar and marble n is taken out. The > question is: How many marbles are in the jar at t=1? >Here is my take on the whole Ross-Littlewood issue. If all the marbles are > labeled as described above, they should all be out of the jar at the end of > the supertask. So I agree with your antagonists on this point. But if they > were labeled differently that need not be the case. Suppose at t=0 ten > marbles are placed in the jar but they are labeled > (1,2,3,4,5,w,w+1,w+2,w+4,w+5) and at t=1/2 ten more marbles are placed in > the jar labeled (6,7,8,9,10,w+6,w+7,w+8,w+9,w+10) etc and each marble > labeled with the nth natural number is removed at time t=1-(1/2)^n. At the > end of the supertask all marbles labeled with natural numbers will be out of > the jar and all the marbles labeled with transfinite ordinal numbers will be > left in the jar. In that case there is an infinite number of marbles in the > jar at the end of the supertask but not one of them is labeled with a > natural number. > Sure, or you can add balls 1-10 and remove 1, then add 11-20 and remove 11, etc, and end up with an infinite number of naturally numbered balls. The question is, if you want a good joke question, can we do it this way, and then change all the labels on the balls at noon, and make them disappear? I mean, if the labels are so important, and we're able to do these supertasks, why can't we make them disappear simply by changing labels? Ultimately, the labels are a distraction, unnecessary for determining that sum(1->oo: 9) diverges. >Some related links: >http://en.wikipedia.org/wiki/Supertask >http://plato.stanford.edu/entries/spacetime-supertasks/ >http://en.wikipedia.org/wiki/Category:Supertasks >-R === Subject: Re: An uncountable countable set On Fri, 27 Oct 2006 00:07:16 -0400, Tony Orlow > I share your and Godel's concerns about point set theory >> >> Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says >> such things, I share Godel's concerns about point set theory. Too bad >> Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's >> concerns, or mathematical logic, or set theory, or point set topology, >> or topology. >> >> MoeBlee >> Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) Of course a little levity, like motions to adjourn, is always in order, Tony. Mathematikers take all this drollery way too seriously. They get all huffy and self righteous when forced to actually explain ~v~~ === Subject: Re: An uncountable countable set > On Fri, 27 Oct 2006 00:07:16 -0400, Tony Orlow > I share your and Godel's concerns about point set theory > Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says > such things, I share Godel's concerns about point set theory. Too bad > Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's > concerns, or mathematical logic, or set theory, or point set topology, > or topology. >> MoeBlee > Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) >Of course a little levity, like motions to adjourn, is always in > order, Tony. Mathematikers take all this drollery way too seriously. > They get all huffy and self righteous when forced to actually explain >~v~~ Smiles, Tony === Subject: Re: An uncountable countable set >> I share your and Godel's concerns about point set theory >Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says > such things, I share Godel's concerns about point set theory. Too bad > Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's > concerns, or mathematical logic, or set theory, or point set topology, > or topology. >MoeBlee Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) To is clearly all cracked up, but why does he blame it on anyone else? === Subject: Re: An uncountable countable set >> I share your and Godel's concerns about point set theory >> >> Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says >> such things, I share Godel's concerns about point set theory. Too bad >> Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's >> concerns, or mathematical logic, or set theory, or point set topology, >> or topology. >> >> MoeBlee >> >> >> Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) To is clearly all cracked up, but why does he blame it on anyone else? Unlike Stephen Tony just likes to give credit where credit is due. ~v~~ === Subject: Re: An uncountable countable set >I share your and Godel's concerns about point set theory >> Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says >> such things, I share Godel's concerns about point set theory. Too bad >> Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's >> concerns, or mathematical logic, or set theory, or point set topology, >> or topology. >> MoeBlee > Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) >> To is clearly all cracked up, but why does he blame it on anyone else? >Unlike Stephen Tony just likes to give credit where credit is due. >~v~~ Yes, Lester, your retorts ar very witty, and sometimes quite to the point, but not usually. But, always sharp. I like that. It was Epistemology 201: The Science of Science, that sucked me into UseNet, way back when. Have we broken that record in this thread yet? Records all fall, eventually. Smiles, Tony === Subject: Re: An uncountable countable set <453d9a1d$1@news2.lightlink> <453e4a3f@news2.lightlink> <453fac14@news2.lightlink> <4540e2bd@news2.lightlink> <454185e7@news2.lightlink> <4542a180@news2.lightlink Yes, Lester, your retorts ar very witty, and sometimes quite to the > point, but not usually. But, always sharp. Not only is Orlow oblivious to logic and ignorant of mathematics, but he's also prone to mistake the dull grunts of the likes of Lester Zick for wit. MoeBlee === Subject: Re: An uncountable countable set >> < endless reiterations of the following >> The only question is According to the rules set up in the problem, is > each ball which is inserted into the vase before noon also removed from > the vase before noon? >> An affirmative answer confirms that the vase is empty at noon. > A negative answer violates the conditions of the problem. > > Which answer does TO choose? >> God, are you a broken record, or what? Let's take this very slowly. Ready? >> Each ball inserted before noon is removed before noon, but at each time >> before noon when a ball is removed, 10 balls have been added, and 9/10 >> of the balls inserted remain. Therefore, at no time before noon is the >> vase empty. Agreed? >> Events including insertions and removals only occur at times t of the >> form t=-1/n, where n e N. Where noon means t=0, there is no t such that >> -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? >> Balls can only leave the vase by removal, each of which must occur at >> some t=-1/n. The vase can only become empty if balls leave. Therefore >> the vase cannot become empty at noon. Agreed? > Not so fast. What do become empty or become empty at mean? >> Not so fast???? We've been laboring this point endlessly. The vase >> goes from a state of balledness to a state of balllessness starting at >> time 0. >Agreed. > >> Balls have to have been removed for this transition to occur. >Yes, but they don't have to have been removed at time 0. In order for emptiness to occur at that time, removals have to occur at that time, if removals are what causes the emptiness. Was that too fast? > >> It is not empty, and it does not become empty, then it is still not >> empty. Agreed? >> When you bring t=0 into the experiment, if anything DOES occur at that >> moment, then the index n of any ball removed at that point must satisfy >> t=-1/n=0, which means that n must be infinite. So, if noon comes, you >> will have balls, but not finitely numbered balls. In this experiment, >> however, t=0 is excluded by the fact that n e N, so noon is implicitly >> impossible to begin with. > === Subject: Re: An uncountable countable set > >> < endless reiterations of the following >> The only question is According to the rules set up in the problem, is > each ball which is inserted into the vase before noon also removed from > the vase before noon? >> An affirmative answer confirms that the vase is empty at noon. > A negative answer violates the conditions of the problem. > > Which answer does TO choose? >> God, are you a broken record, or what? Let's take this very slowly. >> Ready? >> Each ball inserted before noon is removed before noon, but at each time >> before noon when a ball is removed, 10 balls have been added, and 9/10 >> of the balls inserted remain. Therefore, at no time before noon is the >> vase empty. Agreed? >> Events including insertions and removals only occur at times t of the >> form t=-1/n, where n e N. Where noon means t=0, there is no t such that >> -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? >> Balls can only leave the vase by removal, each of which must occur at >> some t=-1/n. The vase can only become empty if balls leave. Therefore >> the vase cannot become empty at noon. Agreed? > Not so fast. What do become empty or become empty at mean? >> Not so fast???? We've been laboring this point endlessly. The vase >> goes from a state of balledness to a state of balllessness starting at >> time 0. > Agreed. > Balls have to have been removed for this transition to occur. > Yes, but they don't have to have been removed at time 0. >> In order for emptiness to occur at that time, removals have to occur at >> that time, if removals are what causes the emptiness. Was that too fast? >In order to have emptiness at noon, all removals must take place no > later than noon, which they are forced to do by the rules of the problem. That means either before noon, or at noon. No balls are removed at noon. Balls remain at every time before noon. You're busted. === Subject: Re: An uncountable countable set >In order to have emptiness at noon, all removals must take place no > later than noon, which they are forced to do by the rules of the problem. >That means either before noon, or at noon. No balls are removed at noon. > Balls remain at every time before noon. You're busted. Except that every ball inserted before noon has been removed before noon according to the specifications of the gedankenexperiment. So it is TO who is busted for violating the rules of the gedankenexperiment. === Subject: Re: An uncountable countable set In order to have emptiness at noon, all removals must take place no > later than noon, which they are forced to do by the rules of the problem. >> That means either before noon, or at noon. No balls are removed at noon. >> Balls remain at every time before noon. You're busted. >Except that every ball inserted before noon has been removed before > noon according to the specifications of the gedankenexperiment. >So it is TO who is busted for violating the rules of the > gedankenexperiment. At least you're not drenching my face with pepper spray. But, then again, I'm not the one who squealed like a piggy. :) Chuckles with blood splatters, Tony === Subject: Re: An uncountable countable set In order to have emptiness at noon, all removals must take place no > later than noon, which they are forced to do by the rules of the problem. >> That means either before noon, or at noon. No balls are removed at noon. >> Balls remain at every time before noon. You're busted. >Except that every ball inserted before noon has been removed before > noon according to the specifications of the gedankenexperiment. >So it is TO who is busted for violating the rules of the > gedankenexperiment. Chuckles with blood splatters, Is TO trying to bring Halloween into this? It doesn't fly. The only relevant question is According to the rules set up in the problem, is each ball which is inserted into the vase before noon also removed from the vase before noon? An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? === Subject: Re: An uncountable countable set >> So, David, you think the fact that balls leave the vase only by being >> removed one at a time, and the fact that at all times before noon there >> are balls in the vase, and the fact that at noon there are no balls in >> the vase, is consistent with the fact that no balls are removed at noon? >> How can you not see the logical inconsistency of an infinitude of balls >> disappearing, not just in a moment, but at no possible moment? Are you >> so steeped in set theory that you cannot see that an unending sequence >> of +10-1 amounts to an unending series of +9's which diverges? What is >> illogical about that? >> In your set-theoretic interpretation of the experiment there is a >> problem which makes your conclusion incompatible with conclusions drawn >> from infinite series, and other basic logical approaches. > I gave a Freshman Calculus interpretation/translation of the problem (no > set theory required). Here is a suitable version: >> For n = 1,2,..., define >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? >> I suppose you either disagree with this interpretation/translation or > you disagree that for this interpretatin V(0) = 0. Which is it? >> t=0 is precluded by n e N and t(n) = -1/n. >Sorry, I don't follow. Were you answering my question? I gave you a > choice: >1. Disagree with the interpretation/translation > 2. Agree with the interpretation/translation, but disagree that V(0) = 0 >Are you picking #1 or #2? I'll choose #2 on the grounds that 0 does not exist in the experiment and that V(0) is therefore without meaning. >Given my interpretation/translation of the problem into Mathematics (see > above) and given that the moment the vase becomes empty means the > first time t >= -1 that V(t) is zero, then it follows that the vase > becomes empty at t = 0 (i.e., noon). >> Yes, now, when nothing occurs at noon, and no balls are removed, what >> else causes the vase to become empty? >No balls are added or removed at noon, but the vase becomes empty at > noon. Through some other mechanism than ball removal? >If you consider the vase becoming empty to be something rather than > nothing, then it is not true that nothing occurs at noon. If by > nothing occurs at noon, you mean no balls are added or removed, then > it is true that nohting occurs at noon. And, if no balls are moved at noon, what causes the vase to become empty at noon? Evaporation? A black hole? >The cause of the vase becoming empty at noon is that all balls are > removed before noon, but at all times between one minute before noon and > noon, there are balls in the vase. The fact that there are balls at all times before noon and that no balls are removed at noon imply that there are balls in the vase at noon, if it exists in the experiment at all to begin with. >Let me ask you the same question regarding the following problem. >Problem: For n = 1,2,..., let > A_n = -1/floor((n+9)/10), > R_n = -1/n. >For n = 1,2,..., define a function B_n by > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >Let V(t) = sum_n B_n(t). What is V(0)? Answer: V(0) = 0. >Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) > is approaching infinity as t approaches zero from the left, what causes > V(0) to be zero? > The fact that you have no upper bound to the naturals. This is the same technique, essentially, which equates the naturals with, say, the evens, or squares of naturals, even though those are proper subsets of the naturals. You can draw a 1-1 correspondence between the balls in and out, sure. There's a bijection there. Infinite bijections do not given any notion of measure unless they are parameterized. Here, you can look at number of balls in the vase as a function of n or of t. In either case, the sum diverges. It is only in trying to consider the unbounded set as completed that you come to this silly conclusion. === Subject: Re: An uncountable countable set >> So, David, you think the fact that balls leave the vase only by being >> removed one at a time, and the fact that at all times before noon there >> are balls in the vase, and the fact that at noon there are no balls in >> the vase, is consistent with the fact that no balls are removed at noon? >> How can you not see the logical inconsistency of an infinitude of balls >> disappearing, not just in a moment, but at no possible moment? Are you >> so steeped in set theory that you cannot see that an unending sequence >> of +10-1 amounts to an unending series of +9's which diverges? What is >> illogical about that? >> In your set-theoretic interpretation of the experiment there is a >> problem which makes your conclusion incompatible with conclusions drawn >> from infinite series, and other basic logical approaches. > I gave a Freshman Calculus interpretation/translation of the problem (no > set theory required). Here is a suitable version: >> For n = 1,2,..., define >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? >> I suppose you either disagree with this interpretation/translation or > you disagree that for this interpretatin V(0) = 0. Which is it? >> t=0 is precluded by n e N and t(n) = -1/n. > Sorry, I don't follow. Were you answering my question? I gave you a > choice: >> 1. Disagree with the interpretation/translation > 2. Agree with the interpretation/translation, but disagree that V(0) = 0 >> Are you picking #1 or #2? >> I'll choose #2 on the grounds that 0 does not exist in the experiment >> and that V(0) is therefore without meaning. > Given my interpretation/translation of the problem into Mathematics (see > above) and given that the moment the vase becomes empty means the > first time t >= -1 that V(t) is zero, then it follows that the vase > becomes empty at t = 0 (i.e., noon). >> Yes, now, when nothing occurs at noon, and no balls are removed, what >> else causes the vase to become empty? > No balls are added or removed at noon, but the vase becomes empty at > noon. >> Through some other mechanism than ball removal? > If you consider the vase becoming empty to be something rather than > nothing, then it is not true that nothing occurs at noon. If by > nothing occurs at noon, you mean no balls are added or removed, then > it is true that nohting occurs at noon. >> And, if no balls are moved at noon, what causes the vase to become empty >> at noon? Evaporation? A black hole? > The cause of the vase becoming empty at noon is that all balls are > removed before noon, but at all times between one minute before noon and > noon, there are balls in the vase. >> The fact that there are balls at all times before noon and that no balls >> are removed at noon imply that there are balls in the vase at noon, if >> it exists in the experiment at all to begin with. > Let me ask you the same question regarding the following problem. >> Problem: For n = 1,2,..., let >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Let V(t) = sum_n B_n(t). What is V(0)? Answer: V(0) = 0. >> Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) > is approaching infinity as t approaches zero from the left, what causes > V(0) to be zero? >> The fact that you have no upper bound to the naturals. This is the same >> technique, essentially, which equates the naturals with, say, the evens, >> or squares of naturals, even though those are proper subsets of the >> naturals. You can draw a 1-1 correspondence between the balls in and >> out, sure. There's a bijection there. Infinite bijections do not given >> any notion of measure unless they are parameterized. Here, you can look >> at number of balls in the vase as a function of n or of t. In either >> case, the sum diverges. It is only in trying to consider the unbounded >> set as completed that you come to this silly conclusion. >Let's see if I understand what you are saying. Consider this math: >-------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. >For n = 1,2,..., define a function B_n: R -> R by > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >Are you saying that V(0) is not equal to zero? > (sigh) I already answered this. Are you just trying to test for consistency in my statements? That gets a little tiresome. I am saying 0 doesn't happen in this experiment. All events are before 0. Any events occurring at 0, by the constraints of the experiment, must have an index n in the sequence such that 1/n=0, but this n cannot exist in the sequence of finite natural numbers. Therefore, nothing happens at t=0. If it did, all finite balls would be gone, but they would be replaced by an uncountable number of infinitely-numbered balls. At the time of each and every event before 0, without exception, more balls are left in the vase than were there before the event, and during (-1,0), the vase is never empty. Combine these two facts, and you get that the vase did not become empty, since nothing happens at noon to the balls to change the state of the vase, and the desired state does not occur before noon. It still has balls. And including noon doesn't change that, but only pushes the potential, countable infinity to the actual, uncountable point. Tony === Subject: Re: An uncountable countable set > I gave a Freshman Calculus interpretation/translation of the problem (no > set theory required). Here is a suitable version: >> For n = 1,2,..., define >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? >> I suppose you either disagree with this interpretation/translation or > you disagree that for this interpretatin V(0) = 0. Which is it? >> t=0 is precluded by n e N and t(n) = -1/n. > Sorry, I don't follow. Were you answering my question? I gave you a > choice: >> 1. Disagree with the interpretation/translation > 2. Agree with the interpretation/translation, but disagree that V(0) = 0 >> Are you picking #1 or #2? >> I'll choose #2 on the grounds that 0 does not exist in the experiment >> and that V(0) is therefore without meaning. > Given my interpretation/translation of the problem into Mathematics (see > above) and given that the moment the vase becomes empty means the > first time t >= -1 that V(t) is zero, then it follows that the vase > becomes empty at t = 0 (i.e., noon). >> Yes, now, when nothing occurs at noon, and no balls are removed, what >> else causes the vase to become empty? > No balls are added or removed at noon, but the vase becomes empty at > noon. >> Through some other mechanism than ball removal? > If you consider the vase becoming empty to be something rather than > nothing, then it is not true that nothing occurs at noon. If by > nothing occurs at noon, you mean no balls are added or removed, then > it is true that nohting occurs at noon. >> And, if no balls are moved at noon, what causes the vase to become empty >> at noon? Evaporation? A black hole? > The cause of the vase becoming empty at noon is that all balls are > removed before noon, but at all times between one minute before noon and > noon, there are balls in the vase. >> The fact that there are balls at all times before noon and that no balls >> are removed at noon imply that there are balls in the vase at noon, if >> it exists in the experiment at all to begin with. > Let me ask you the same question regarding the following problem. >> Problem: For n = 1,2,..., let >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Let V(t) = sum_n B_n(t). What is V(0)? Answer: V(0) = 0. >> Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) > is approaching infinity as t approaches zero from the left, what causes > V(0) to be zero? >> The fact that you have no upper bound to the naturals. This is the same >> technique, essentially, which equates the naturals with, say, the evens, >> or squares of naturals, even though those are proper subsets of the >> naturals. You can draw a 1-1 correspondence between the balls in and >> out, sure. There's a bijection there. Infinite bijections do not given >> any notion of measure unless they are parameterized. Here, you can look >> at number of balls in the vase as a function of n or of t. In either >> case, the sum diverges. It is only in trying to consider the unbounded >> set as completed that you come to this silly conclusion. > Let's see if I understand what you are saying. Consider this math: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. >> For n = 1,2,..., define a function B_n: R -> R by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Are you saying that V(0) is not equal to zero? >> (sigh) I already answered this. Are you just trying to test for >> consistency in my statements? That gets a little tiresome. >Sorry, but I think you answered a different question than I asked. > >> I am saying 0 doesn't happen in this experiment. All events are before >> 0. Any events occurring at 0, by the constraints of the experiment, must >> have an index n in the sequence such that 1/n=0, but this n cannot exist >> in the sequence of finite natural numbers. Therefore, nothing happens at >> t=0. If it did, all finite balls would be gone, but they would be >> replaced by an uncountable number of infinitely-numbered balls. >> At the time of each and every event before 0, without exception, more >> balls are left in the vase than were there before the event, and during >> (-1,0), the vase is never empty. >> Combine these two facts, and you get that the vase did not become empty, >> since nothing happens at noon to the balls to change the state of the >> vase, and the desired state does not occur before noon. It still has >> balls. And including noon doesn't change that, but only pushes the >> potential, countable infinity to the actual, uncountable point. >You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >-------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? > On the surface, you math appears correct, but that doesn't mend the obvious contradiction in having an event occur in a time continuum without occupying at least one moment. It doesn't explain how a divergent sum converges to 0. Basically, what you prove, if V(0)=0, is that all finite naturals are removed by noon. I never disagreed with that. However, to actually reach noon requires infinite naturals. Sure, if V is defined as the sum of all finite balls, V(0)=0. But, I've already said that, several times, haven't I? Isn't that an answer to your question? === Subject: Re: An uncountable countable set >On the surface, you math appears correct, but that doesn't mend the > obvious contradiction in having an event occur in a time continuum > without occupying at least one moment. It doesn't explain how a > divergent sum converges to 0. Basically, what you prove, if V(0)=0, is > that all finite naturals are removed by noon. I never disagreed with > that. However, to actually reach noon requires infinite naturals. Where do these alleged infinite naturals come from? Then are certainly not available in the original gedankenexperiment, which takes place in a mathematical world compatible with ZF or NBG. They spring full-blown from TO's intuition, which is hostile to all standard mathematics, and so irrelevant to all standard mathematics. And the gedankenexperiment occurs in standard mathematics. === Subject: Re: An uncountable countable set > >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. > Where do these alleged infinite naturals come from? Then are certainly > not available in the original gedankenexperiment, which takes place in a > mathematical world compatible with ZF or NBG. They come from t=-1/n and t=0, thus 0=-1/n. Idiot. >They spring full-blown from TO's intuition, which is hostile to all > standard mathematics, and so irrelevant to all standard mathematics. >And the gedankenexperiment occurs in standard mathematics. t=-1/n ^ t=0 -> -1/n=0. T v F? === Subject: Re: An uncountable countable set > And the gedankenexperiment occurs in standard mathematics. >t=-1/n ^ t=0 -> -1/n=0. T v F? t=-1/n ^ t=0 -> 1 = 2 is just as true as t=-1/n ^ t=0 -> -1/n = 0 === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >-------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >On the surface, you math appears correct, but that doesn't mend the > obvious contradiction in having an event occur in a time continuum > without occupying at least one moment. It doesn't explain how a > divergent sum converges to 0. Basically, what you prove, if V(0)=0, is > that all finite naturals are removed by noon. I never disagreed with > that. However, to actually reach noon requires infinite naturals. Sure, > if V is defined as the sum of all finite balls, V(0)=0. But, I've > already said that, several times, haven't I? Isn't that an answer to > your question? I think it is an answer. Just to be sure, please confirm that you agree that, with the definitions above, V(0) = 0. Is that correct? -- David Marcus === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? >I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? > Sure, all finite balls are gone at noon. === Subject: Re: An uncountable countable set > I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? > >Sure, all finite balls are gone at noon. And in any system compatible with ZF or NBG there aren't any others. === Subject: Re: An uncountable countable set I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? > >> Sure, all finite balls are gone at noon. >And in any system compatible with ZF or NBG there aren't any others. When did I claim my ideas were consistent with those theories? === Subject: Re: An uncountable countable set I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? > >> Sure, all finite balls are gone at noon. >And in any system compatible with ZF or NBG there aren't any others. >When did I claim my ideas were consistent with those theories? But we claim that the original gedankenexperiment must be compatible with some standard set theory unless it explicitly specifies some other set theory, and ZF and NBG are the only ones around. If TO claims some other set theory, he has yet to specify which one, so we can also assume that he is referring to one of the standards. === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? >I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? >Sure, all finite balls are gone at noon. Please note that there are no balls or time in the above mathematics problem. However, I'll take your Sure as agreement that V(0) = 0. Let me ask you a question about this mathematics problem. Please answer without using the words balls, vase, time, or noon (since these words do not occur in the problem). First some discussion: For each n, B_n(0) = 0 and B_n is continuous at zero. In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for e < t <= 0. In other words, B_n is not changing near zero. Now, V is the sum of the B_n. As t approaches zero from the left, V(t) grows without bound. In fact, given any large number M, there is an e < 0 such that for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). Now the question: How do you explain the fact that V(t) goes from being very large for t a little less than zero to being zero when t equals zero even though none of the B_n are changing near zero? -- David Marcus === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? > I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? >> Sure, all finite balls are gone at noon. >Please note that there are no balls or time in the above mathematics > problem. However, I'll take your Sure as agreement that V(0) = 0. > Okay. > Let me ask you a question about this mathematics problem. Please answer > without using the words balls, vase, time, or noon (since these > words do not occur in the problem). I'll try. >First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > zero. What??? How do you conclude that anything besides time is continuous at 0, where yo have an ordinal discontinuity???? Please explain. In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > e < t <= 0. There is no e<0 such that e sum of the B_n. As t approaches zero from the left, V(t) grows without > bound. In fact, given any large number M, there is an e < 0 such that > for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). >Now the question: How do you explain the fact that V(t) goes from being > very large for t a little less than zero to being zero when t equals > zero even though none of the B_n are changing near zero? > I'll consider answering that when you correct the errors above. Sorry. === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? > I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? >> Sure, all finite balls are gone at noon. >Please note that there are no balls or time in the above mathematics > problem. However, I'll take your Sure as agreement that V(0) = 0. Okay. >Let me ask you a question about this mathematics problem. Please answer > without using the words balls, vase, time, or noon (since these > words do not occur in the problem). >I'll try. >First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > zero. >What??? How do you conclude that anything besides time is continuous at > 0, where yo have an ordinal discontinuity???? Please explain. I thought we agreed above to not use the word time in discussing this mathematics problem? As for your question, let's look at B_2 (the argument is similar for the other B_n). B_2(t) = 1 if A_2 <= t < R_2, 0 if t < A_2 or t >= R_2. Now, A_2 = -1 and R_2 = -1/2. So, B_2(t) = 1 if -1 <= t < -1/2, 0 if t < -1 or t >= -1/2. In particular, B_2(t) = 0 for t >= -1/2. So, the value of B_2 at zero is zero and the limit as we approach zero is zero. So, B_2 is continuous at zero. > In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > e < t <= 0. >There is no e<0 such that e In other words, B_n is not changing near zero. > Infinitely more quickly but not. That's logical. And wrong. Not sure what you mean. > Now, V is the > sum of the B_n. As t approaches zero from the left, V(t) grows without > bound. In fact, given any large number M, there is an e < 0 such that > for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). >Now the question: How do you explain the fact that V(t) goes from being > very large for t a little less than zero to being zero when t equals > zero even though none of the B_n are changing near zero? >I'll consider answering that when you correct the errors above. Sorry. -- David Marcus === Subject: Re: An uncountable countable set >Now, V is the >sum of the B_n. As t approaches zero from the left, V(t) grows without >bound. In fact, given any large number M, there is an e < 0 such that >for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). Now the question: How do you explain the fact that V(t) goes from being >very large for t a little less than zero to being zero when t equals >zero even though none of the B_n are changing near zero? >I'll consider answering that when you correct the errors above. Sorry. AS there are no errors in David's analysis, but many in TO's, that indicates that TO has no reasonable answers. But that is hardly news. === Subject: Re: An uncountable countable set > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: >> -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R by > > B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. > > Let V(t) = sum_n B_n(t). > -------------------------- >> Just looking at these definitions of sequences and functions from R > (the > real numbers) to R, and assuming that the sum is defined as it would be > > in a Freshman Calculus class, are you saying that V(0) is not equal to > 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? > I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? >> Sure, all finite balls are gone at noon. >Please note that there are no balls or time in the above mathematics > problem. However, I'll take your Sure as agreement that V(0) = 0. Okay. >Let me ask you a question about this mathematics problem. Please answer > without using the words balls, vase, time, or noon (since these > words do not occur in the problem). >I'll try. First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > zero. >What??? How do you conclude that anything besides time is continuous at > 0, where yo have an ordinal discontinuity???? Please explain. For anyone except TO, it would be obvious that each B_n has only two points of discontinuity, A_n and R_n, neither of which occurs at t=0. >In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > e < t <= 0. >There is no e<0 such that eIn other words, B_n is not changing near zero. >Infinitely more quickly but not. That's logical. And wrong. Why is it that, according to TO, everything logical is wrong? > I'll consider answering that when you correct the errors above. As the errors above are all TO's, TO should correct them. === Subject: Re: An uncountable countable set > For n = 1,2,..., suppose we have numbers A_n and R_N (the addition and > removal times of ball n where time is measured in minutes before > noon). For n = 1,2,..., define a function B_n by >> B_n(t) = 1 if A_n <= t < R_n, > 0 if t < A_n or t >= R_n. >> Fine for each ball n. > Let V(t) = sum{n=1}^infty B_n(t). Let L = lim_{t -> 0-} V(t). Let S = > V(0). Let T be the number of balls that you say are in the vase at > noon. >> You are summing B_n(t) to oo? >The sum is over all positive integers. There is no B_oo. I'm sticking to > standard Calculus notation. Does that change your answers below? Not really, but it's hard to tell with that notation whether you are including noon or not. >Problem 1. For n = 1,2,..., define >> A_n = -1/floor((n+9)/10), > R_n = -1/n. >> Then L = infinity, S = 0, and T = undefined. >> I say that if noon exists, there are an infinite number of balls in the >> vase. n=oo -> L=T. >By noon exists do you mean there is a ball B_oo? There isn't. No kidding. That's why noon cannot be part of the experiment. > > Problem 2. For n = 1,2,..., define >> A_n = -1/n, > R_n = -1/(n+1). >> Then L = 1, S = 0, T = 1. >> Yeah, L=T again. > Problem 3. For n = 1,2,..., define >> A_n = -1, > R_n = -1/n. >> Then L = infinity, S = 0, T = 0. >> L=lim(x->oo: oo-x) = 0 <> oo >L is the limit of V(t) as t approaches zero from the left. So, L = oo. >I don't know what you mean by lim(x->oo: oo-x). We don't normally > define things like oo-x, for x an integer, in Calculus. Of course, the > most natural definition would be for oo-x to equal oo, in which case > your limit would also be oo. But, you say it is zero. If oo=oo then oo-oo=0. > >> L=T > Tony, can you give us a general procedure to let us determine T given > the A_n's and B_n's? >> You can keep track of the points between your time vortexes and what's >> going on during those periods, for starters. >I'm afraid I don't know what I'm supposed to keep track of. In truth, I > thought that by calculating V, I was keeping track. But, neither of the > two quantities I can get from V, i.e., L and S, seem to consistently > match your value. > You are supposed to keep in mind the coupling of 10 additions with each subtraction, and note that this sum of balls cannot converge to 0 no matter how long you keep it up. === Subject: Re: An uncountable countable set > Also, supposing for the sake of argument that there are infinitely > number balls, if a ball is added at time -1/(2^floor(n/10)), and removed > at time -1/(2^n)), then the balls added at time t=0, are those > where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic > on these infinite numbers works), so those balls are also removed at noon and > never spend any time in the vase. >> Yes, the insertion/removal schedule instantly becomes infinitely fast in >> a truly uncountable way. The only way to get a handle on it is to >> explicitly state the level of infinity the iterations are allowed to >> achieve at noon. When the iterations are restricted to finite values, >> noon is never reached, but approached as a limit. > Suppose we only do an insertion or removal at t = 1/n for n a natural > number. What do you mean by noon is never reached? >> 1/n>0 > Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. >> But, I don't follow. Translating -1/n < 0 back into words, I get all > insertions and removals are before noon. However, I asked you what > noon is never reached means. Are you saying that noon is never > reached means that all insertions and removals are before noon? >> Yes, David. What else happens in this experiment besides insertions and >> removals of naturals at finite times before noon? If the infinite >> sequence of events is actually allowed to continue until t=0, then you >> are talking about events not indexed with natural numbers, so you're not >> talking about the same experiment. If noon is not allowed, and all times >> in the experiment are finitely before noon, well, at none of those times >> does the vase empty, as we all agree. This is why I am asking when this >> occurs. It can't, given the constraints of the problem. > Does your Yes at the beginning of your reply mean that you agree that > noon is never reached means that all insertions and removals are > before noon. By mean, I mean that that is what the words mean, not > that the two statements are equivalent or deducible from each other. >> Yes, every event, every insertion or removal, happens at a specific time >> before noon. At each of those times, the vase is non-empty. Nothing else >> occurs, as far as insertions of removals. Is that clear enough? So, when >> does the vase become empty, >At noon. When nothing happens? What balls are removed at noon? Can the vase become empty by a means other than ball removal? > >> and how? >By removing all balls before noon, but leaving some balls in the vase at > every time before noon and after one minute before noon. > Uhhhhh......does that actually make sense to you? Can it? There are balls at every moment before noon, nothing happens at noon, but they are all gone suddenly? What causes the vase to be empty at noon, if not removals at noon, which cannot occur? === Subject: Re: An uncountable countable set > But none of Robinson's non-standard numbers are cardinalities. >> No kidding. They actually make sense. >You said you have not properly studied chapter II in the book - the one > that includes mathematical logic, model theory, and set theory (does it > not? I'll stand corrected if it doesn't). What are you going to say > when you find out that what you say makes sense rests on a foundation > of set theory that you say doesn't make sense? Or, if I'm incorrect > that Robinson's work in non-standard analysis doesn't presuppose basic > mathematical logic, model theory, and set theory, then I'll benefit by > being corrected in my admittedly cursory understanding of the matter. >MoeBlee > Uh, if Robinson's thesis is built upon transfinite set theory, then that is evidence right there that it's inconsistent, since you have a smallest infinity, omega, but Robinson has no smallest infinity. Robinson doesn't use ordinals or cardinals that I've seen. He basically defines what a well-formed formula is in his system, which is a little more restrictive that some others, it seems, and uses the language to extend what can be said about finite n in N to include infinite n in *N. TOEKnee === Subject: Re: An uncountable countable set On Thu, 26 Oct 2006 23:28:04 -0400, Tony Orlow [. . .] >> We JUST agreed that 'smallest infinity' means two different things when >> referring to ordinals and when referring to certain kinds of other >> make sure this was clear, and then you agreeed, you NOW come back to >> conflate the two ANYWAY! Ahem. I said that Robinson's analysis seems to have nothing to do with >transfinitology. They appear to be unrelated. However, they cme to two >very different conclusions regarding a basic question: is there a >smallest infinite number? It seems clear to me there is not, for the >very same reason that Robinson uses: if there is an infinite number, you >can subtract 1 and get a different, smaller infinite number. It's the >same logic y'all use to argue that there's no largest finite. It's >correct. The Twilight Zone between finite and infinite CANNOT really be >pinpointed that way. Hey, Tony. You know this is an interesting problem but I think you're wasting your time here arguing the issue with the Holy Order of Self Righteous Mathematikers. Let me outline my own thinking for you. I think there is a smallest infinity but that subtracting finites from infinites isn't the way to get at the problem because as far as I can tell arithmetic operations cannot be defined between infinites and finites any more than finite division by zero can be. Instead you need a different approach altogether and I suspect the way to get at the problem is to assess the kind of infinity according to the number of infinitesimals in various intervals. And in this manner I suspect you'll find the infinity associated with straight line segments is the smallest and various kinds of curves larger. ~v~~ === Subject: Re: An uncountable countable set > On Thu, 26 Oct 2006 23:28:04 -0400, Tony Orlow [. . .] >We JUST agreed that 'smallest infinity' means two different things when > referring to ordinals and when referring to certain kinds of other > make sure this was clear, and then you agreeed, you NOW come back to > conflate the two ANYWAY! >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. They appear to be unrelated. However, they cme to two >> very different conclusions regarding a basic question: is there a >> smallest infinite number? It seems clear to me there is not, for the >> very same reason that Robinson uses: if there is an infinite number, you >> can subtract 1 and get a different, smaller infinite number. It's the >> same logic y'all use to argue that there's no largest finite. It's >> correct. The Twilight Zone between finite and infinite CANNOT really be >> pinpointed that way. >Hey, Tony. You know this is an interesting problem but I think you're > wasting your time here arguing the issue with the Holy Order of Self > Righteous Mathematikers. Let me outline my own thinking for you. Alright, but keep in mind that changing the way the world works involves changing people's minds. :) >I think there is a smallest infinity but that subtracting finites from > infinites isn't the way to get at the problem because as far as I can > tell arithmetic operations cannot be defined between infinites and > finites any more than finite division by zero can be. Why do you think there is a smallest infinity, when removing a finite portion thereof leaves a, smaller, infinite portion? >Instead you need a different approach altogether and I suspect the way > to get at the problem is to assess the kind of infinity according to > the number of infinitesimals in various intervals. And in this manner > I suspect you'll find the infinity associated with straight line > segments is the smallest and various kinds of curves larger. >~v~~ You are intuiting in the derivative sense. As segments get smaller, on whatever curve, the angle between them decreases. The trick here is to declare, as Ross aludes, to a universe, a complete range of values for the set or sets, and measure according to that range. I have a feeling maybe you'll get it soon. It fits, I'm sure, with some of your ideas. You do have ideas, don't you? ;) 01oo === Subject: Re: An uncountable countable set <45310552@news2.lightlink> <45343a7a@news2.lightlink> <45358b63@news2.lightlink> <4539117c@news2.lightlink> <453b39cb@news2.lightlink> <453d16f3@news2.lightlink> <453d577b@news2.lightlink> <4540d449@news2.lightlink> <45417cb7@news2.lightlink Ahem. I said that Robinson's analysis seems to have nothing to do with > transfinitology. You agreed it is not an ordering by cardinality. > They appear to be unrelated. However, they cme to two > very different conclusions regarding a basic question: is there a > smallest infinite number? It seems clear to me there is not, for the > very same reason that Robinson uses: if there is an infinite number, you > can subtract 1 and get a different, smaller infinite number. It's the > same logic y'all use to argue that there's no largest finite. It's > correct. The Twilight Zone between finite and infinite CANNOT really be > pinpointed that way. So, that's a verrry basic discrepancy. There is clearly a contradiction > between the two theories. They can't both be right about that, can they? > Is there, and at the same time is not, a smallest infinite number? YOU ARE NOT LISTENING. They TWO DIFFERENT ORDERINGS. And the use of the word 'infinite' is A DIFFERENT SENSE in the two different contexts. So there is NO CONTRADICTION. The reasons you don't get any of this are (1) you don't read the material except to look for quotes and OUT OF CONTEXT passages that you think you can use to bolster your own nonsense. You don't learn the mathematical logic and set theory that are the basis for the material and even pretty much ignore the mathemtatical logic and set theory that the author himself summarizes in the book. (You need to start by learning how to work in the predicate calculus), and (2) you don't listen when someone tries to warn you about the confusions you are making due to your not understanding the basis and context. > No, it is NOT a contradiction with set theory and there being a > smallest infinite ordinal and smallest infinite cardinal that there are > also non-standard orderings (which are NOT cardinality or ordinal > ordering, as even YOU recognized) that have what are CALLED 'infinite > elements' but with no least one. In transfinitology? Why would anyone argue against me saying there is no > smallest infinite? Sorry, you're not meshing. Because, AGAIN, for the TENTH TIME, 'smallest infinite' regarding ordinals and cardinals is a DIFFERENT SUBJECT from 'smallest infinite' regarding elements in certain non-standard models. > Eat me. Do you maintain that the two theories are compatible with each > other? Is there, and also not, a smallest infinity. They're not in conflict, becuase 'smallest infinite' means something DIFFERENT in the different contexts. How many times will I say that while you STILL refuse to hear it? > Geeze, calm down. I am not conflating them as if they were the same > thing. I am clearly stating that they are obviously mutually > incompatible, No, that IS conflating them by saying there is an incompatiblity. I would agree that the terminology is unfortunate, since there are two different senses of 'infinite' in play. But were we to fully formalize, we would have predicate symbols, not English language words, so we wouldn't have this problem. But anyone who knows anything about the subject can understand, even without such formalization, that 'infinite' is a WORD that is being used in DIFFERENT senses and there is no incompatiblitiy in the mathematics if one just recognizes the DIFFERENT senses that are being used. > You need to relax. I worked on the first chapter for a bit, and got most > of the way through, but started to get bogged down, so I skipped ahead > to see what the next chapter held, and it did. So, sue me. At least I > didn't run to the Cliff Notes... You got bogged down because that chapter is a graduate level SUMMARY of material that most readers of that book will have already digested before even picking up that book. You need to learn the basics of mathematical logic and set theory to even be able to very much benefit from such a summary of material as in that book that is even more advanced in mathematical logic than the very basic mathematical logic you still have not learned. > Well, at least I'm not being such an obnoxious jerk. Or, maybe you think > I am. Yes, you are being a real obnoxious jerk. > Read what Robinson says, and think about it. If you're really so > interested in alternative theories, I'd think you'd go to the source. You're being an obnoxious jerk AGAIN. Of course I want to read that book, along with Martin Davis's book and others on non-standard analysis, and Edward Nelson and internal set theory, and all kinds of things. But unlike you, I lack the arrogance and foolishness to think that I can skip the basics. Therefore, given that I am not a Zeno machine of mathematical learning, I have to put certain reading on hold while I get caught up on certain basics and even on certain advanced material that is related. That is not a reflection of insincerity in my endeavor to know about alternatives, but rather of my sincerity in that endeavor. MoeBlee === Subject: Re: An uncountable countable set >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. >You agreed it is not an ordering by cardinality. Yes, and you asked me what I would think if it was derived from transfinite set theory. You're tripping over your tail. > >> They appear to be unrelated. However, they cme to two >> very different conclusions regarding a basic question: is there a >> smallest infinite number? It seems clear to me there is not, for the >> very same reason that Robinson uses: if there is an infinite number, you >> can subtract 1 and get a different, smaller infinite number. It's the >> same logic y'all use to argue that there's no largest finite. It's >> correct. The Twilight Zone between finite and infinite CANNOT really be >> pinpointed that way. >> So, that's a verrry basic discrepancy. There is clearly a contradiction >> between the two theories. They can't both be right about that, can they? >> Is there, and at the same time is not, a smallest infinite number? >YOU ARE NOT LISTENING. WHAT???????? I CAN'T HEAR YOU THROUGH THE FOG!!!!! They TWO DIFFERENT ORDERINGS. I think you meant to insert a be's in there somewhere. And the use of the > word 'infinite' is A DIFFERENT SENSE in the two different contexts. So > there is NO CONTRADICTION. OHHHHH!!!!!!! WOW!!!!! NO CONTRADICTION!!!!! WITHIN MATHEMATICS!!! THAT'S GREAT!!!!! WAS THAT THE PHONE????? >The reasons you don't get any of this are (1) you don't read the > material except to look for quotes and OUT OF CONTEXT passages that you > think you can use to bolster your own nonsense. Read Robinson for real and then comment. See how quickly you can absorb his actual section on logical tools, and whether you might skip ahead and then back. You don't learn the > mathematical logic and set theory that are the basis for the material > and even pretty much ignore the mathemtatical logic and set theory that > the author himself summarizes in the book. (You need to start by > learning how to work in the predicate calculus), and (2) you don't > listen when someone tries to warn you about the confusions you are > making due to your not understanding the basis and context. > Well, it doesn't help when they freak out because they can't handle questioning certain iffy logical constructs. You're getting a tad edgy lately, which I understand. I'm not floating on a cloud myself. Life is. > No, it is NOT a contradiction with set theory and there being a > smallest infinite ordinal and smallest infinite cardinal that there are > also non-standard orderings (which are NOT cardinality or ordinal > ordering, as even YOU recognized) that have what are CALLED 'infinite > elements' but with no least one. >> In transfinitology? Why would anyone argue against me saying there is no >> smallest infinite? Sorry, you're not meshing. >Because, AGAIN, for the TENTH TIME, 'smallest infinite' regarding > ordinals and cardinals is a DIFFERENT SUBJECT from 'smallest infinite' > regarding elements in certain non-standard models. > Why does the very mention of no smallest infinite elicit such bile, then? Why is this theiry so defensive? Maybe because it sucks. It's like the Fonzie of mathematical theories. Holy Cannoli, that's exactly what it is. Ayyyyy!!!! Who wants to ride on my bike, babe? >> Eat me. Do you maintain that the two theories are compatible with each >> other? Is there, and also not, a smallest infinity. >They're not in conflict, becuase 'smallest infinite' means something > DIFFERENT in the different contexts. How many times will I say that > while you STILL refuse to hear it? > So, either smallest has two meanings, or infinite has tow meanings, or both. Would you like to elucidate the matter by enumerating the various definitions of small and infinite? A table might be nice... >> Geeze, calm down. I am not conflating them as if they were the same >> thing. I am clearly stating that they are obviously mutually >> incompatible, >No, that IS conflating them by saying there is an incompatiblity. I thought conflating was more like making one of them as important as the other, when it's not. Let's see..... to fuse into one entity; merge: to conflate dissenting voices into one protest. - dictionary Well, I wonder if smallest infinity is one entity or two... >I would agree that the terminology is unfortunate, since there are two > different senses of 'infinite' in play. Two? Which? Countable and uncountable? More than one uncountable and therefore more than two? Is that even decidable in a theory which doesn't use measure? But were we to fully formalize, > we would have predicate symbols, not English language words, so we > wouldn't have this problem. Words are not the problem. A symbol CAN be an infinite design, with uncountably many pieces, if desired. It's just not usually necessary. But anyone who knows anything about the > subject can understand, even without such formalization, that > 'infinite' is a WORD that is being used in DIFFERENT senses and there > is no incompatiblitiy in the mathematics if one just recognizes the > DIFFERENT senses that are being used. > Oh, so it's okay for me to say the set of naturals in not infinite in MY sense of the word? Great. Remember that, and pass it around. Time's a changin', Boss Man. Times O' Flux. >> You need to relax. I worked on the first chapter for a bit, and got most >> of the way through, but started to get bogged down, so I skipped ahead >> to see what the next chapter held, and it did. So, sue me. At least I >> didn't run to the Cliff Notes... >You got bogged down because that chapter is a graduate level SUMMARY > of material that most readers of that book will have already digested > before even picking up that book. Yes, no kidding. There were references to terms and definitions which I could at first glean fairly well, but as they built upon each other, the references got buried upon each other, and I lost track. I have never been a graduate mathematics student. Pardon me. I am not sure that works to my disadvantage in this particular instance. My shackles are non-mathematical. :) This is why, when I choose a thread and participate, it takes forever to end. I am looking in every direction, not to the front of the lecture hall, to find the exotic critters under each rock and in each log. There are underlying logical questions which you and others consider settled, but then again, so do believers, in matters pertaining to God. It's just that, these days, no one believes in math. Except for us cranks. You know, the ones that live in the universe. We seek consistency in thought. That search is not over. Look at the universe. It would be fun to trade lives with you, for about a week. Then I'd have to make your cat catch its own mouse. :) You need to learn the basics of > mathematical logic and set theory to even be able to very much benefit > from such a summary of material as in that book that is even more > advanced in mathematical logic than the very basic mathematical logic > you still have not learned. (sigh) You mean I have not been successfully indoctrinated. That is very true. I remain mathematically feral. {8D }Bk > >> Well, at least I'm not being such an obnoxious jerk. Or, maybe you think >> I am. >Yes, you are being a real obnoxious jerk. I didn't realize. My apologies. Oh wait. That was after you started. Yeah, now I'm being an obnoxious jerk. Sto correctato. > >> Read what Robinson says, and think about it. If you're really so >> interested in alternative theories, I'd think you'd go to the source. >You're being an obnoxious jerk AGAIN. I think you mean, still? Of course I want to read that > book, along with Martin Davis's book and others on non-standard > analysis, and Edward Nelson and internal set theory, and all kinds of > things. Of course. And Check out George Boole's An Investigation into the Laws of Thought. What do you think of the laws of Boolean logic being derived logically from x^2=x? Check it, Boyeeee!!! In Da House!!! Yo George. And, no, I don't mean Georg. :) OK, I'll stop. :) But unlike you, I lack the arrogance and foolishness to think > that I can skip the basics. Then perhaps your arrogance manifests itself in other ways. You might want to think about that. Therefore, given that I am not a Zeno > machine of mathematical learning, I have to put certain reading on hold > while I get caught up on certain basics and even on certain advanced > material that is related. Welcome to the club, and while you're getting defensive about not being the repository of all human or other knowledge, you might want to consider, as I do while seeing othersget bent out of shape, that it might be due to real world concerns, distractions, and needs. I shouldn't even be wasting my time on this, by any normal sensibilities, given what's been and remains on my plate. Maybe that's true of you. Or maybe you have no excuse at all. But I suspect you do. So, take you time. But do check out Non-Standard Analysis, when you get the time. I'm doing Boole and Robinson. We all choose our paths. That is not a reflection of insincerity in my > endeavor to know about alternatives, but rather of my sincerity in that > endeavor. >MoeBlee > Moe, I don't question your sincerity at all. I do believe you are seeking as you say you are. But, I do question your appetite for deduction, and recommend more questions about laying the foundational assumptions beforehand, inductively. My feeling is that questioning of basic truths is too rare, and I think you're up to it. So, I invite you on the hunt. Virgil aside, there are minds here with which to compare intuitive pictures which lead to the next wave. My guess is that a riff will mean something to you soon. Have a toke and close your eyes, and watch what you see. Oh, and read Robinson. It's heady. :) Peace, Tony === Subject: Re: An uncountable countable set <45310552@news2.lightlink> <45343a7a@news2.lightlink> <45358b63@news2.lightlink> <4539117c@news2.lightlink> <453b39cb@news2.lightlink> <453d16f3@news2.lightlink> <453d577b@news2.lightlink> <4540d449@news2.lightlink> <45417cb7@news2.lightlink> <454298b7@news2.lightlink >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. > You agreed it is not an ordering by cardinality. Yes, and you asked me what I would think if it was derived from > transfinite set theory. You're tripping over your tail. So what? You agreed that it is not a cardinality ordering. And so the fact that there is no least infinite (in a different sense of 'infinite') in one ordering, which is not a cardinality ordering, is not a contradiction or incompatiblity with there being a least infinite in a cardinality ordering. > Read Robinson for real and then comment. See how quickly you can absorb > his actual section on logical tools, and whether you might skip ahead > and then back. That's the point. You can't easily (or even at all) absorb that material in the chapter on logic without having studied the basic material that is in undergraduate logic texts. > You don't learn the > mathematical logic and set theory that are the basis for the material > and even pretty much ignore the mathemtatical logic and set theory that > the author himself summarizes in the book. (You need to start by > learning how to work in the predicate calculus), and (2) you don't > listen when someone tries to warn you about the confusions you are > making due to your not understanding the basis and context. >Well, it doesn't help when they freak out because they can't handle > questioning certain iffy logical constructs. You're getting a tad edgy > lately, which I understand. I'm not floating on a cloud myself. Life is. It's got nothing to do with me questioning logical constructs. The problem is that you spout about non-standard analysis while you haven't even BEGUN to address the rudiments of the subject matter, while you instead SKIP the rudiments without which an understanding of the more advanced material cannot be had. > Because, AGAIN, for the TENTH TIME, 'smallest infinite' regarding > ordinals and cardinals is a DIFFERENT SUBJECT from 'smallest infinite' > regarding elements in certain non-standard models. >Why does the very mention of no smallest infinite elicit such bile, > then? Whose bile? I'm not rankled by there being no least infinite member of certain domains (again, that is not a cardinality 'infinite') and wouldn't even be put off by an alternative theory that somehow had no least infinite cardinality. But what does irk me is that after I cleared the way for you NOT to conflate two different contexts of 'least infinite', and after YOU even granted the difference, you conflated them anyway, thus to spout nonsense on the Internet yet again. > So, either smallest has two meanings, or infinite has tow meanings, or > both. Yes, they have at LEAST two meanings, as they are English used in different contexts. A student of the subject will understand the differences in context, but a phony baloney like you will be oblivious to those differences because a poseur like you has no interest in actually studying and understanding the material but rather is just looking for quotes and and bits and pieces of ideas for fodder to put into Internet posts defending his half-baked commentary on a subject of which he knows nothing. > to fuse into one entity; merge: Right, you merged one sense of 'least infinite' with another. > 1 a : to bring together : FUSE b : CONFUSE > 2 : to combine (as two readings of a text) into a composite whole [all caps original] You CONFUSED one sense with another as you failed to distinguish them, as you combined them into one while they are separate. You conflated them. > I would agree that the terminology is unfortunate, since there are two > different senses of 'infinite' in play. Two? Which? Countable and uncountable? How IDIOTIC of you to ask that. I spent about three posts already, using all-caps even, to say that you are conflating the cardinality sense with the sense of points in a certain ordering in the universe of a non-standard model. Not countable and uncountable - those are cardinality. AGAIN, for about the TENTH time: You are confusing 'infinite' in the sense of cardinality with 'infinite', in the very special sense of the context of non-standard analysis, of certain orderings that are not necessarily cardinality orderings. Maybe the word 'position' will help. You are confusing 'infinite' in the sense of cardinality with 'infinite' in the sense of position in certain orderings where we CALL certain members of a universe 'finite members' (though, again, this is a DIFFERENT sense of finite, not necessarily a cardinality sense) and other members of the univers 'infinite' since the infinite members are all related from the left by ALL of the finite members. It has to do with a relation on the universe of the non-standard model. If R is the relation, and S is the set of standard (finite) members of the universe, then the infinite members are the x's such that for any y is S, we have is in R. But R is NOT the relation of 'y is of less cardinality than x'. Also, as to 'countable' and 'uncountable', those are not different senses of 'infinite'. First, 'countable' includes finite. And the difference between 'denumerable' and 'uncountable' (which is, between 'countably infinite' and 'uncountable') is not a matter of different senses of 'infinite' in the way I'm mentioning regarding the difference between cardinality and position in certain orderings of non-standard universes. Rather, the difference betweeen 'denumerable' and 'uncountable' is a difference between infinite sets that are still Your raising 'countable' and 'uncountable' here does not come from you sincerely wanting to understand non-standard analysis or set theory or mathematics, but rather is an irrelevent diversion so that you can continue to argue about this from your position of ignorance. > Oh, so it's okay for me to say the set of naturals in not infinite in MY > sense of the word? Use words however you like. But if you want people to understand you, then you need to set up your context and definitions. And if you have a book, organized and systematic, in which you set up your context and definitions, but people ignore that chapter (such as you for all practical purposes ingore the chapter in the book we're talking about, but more importantly, you ignore the basic body of mathematical logic, set theory, and mathematics that that book uses as that book is written for an audience that has already worked through the basics that the book only summarizes in the logic chapter), then you'll have every right to complain about such sloppy readers of your book as I am complaining about your sloppy reading. Anyway, I said myself that I think the double sense of 'infinite' is unfortunate. Personally, I wish a different word had been chosen for non-standard analysis, and I wish there were more uniformity among authors and mathematicians in the natural language terminology so that readers would not have to be so very mindful of context and overall vagaries of terminology that occur in mathematical writing. However, that being my personal feeling, it is still not not an excuse for your not studying the basics of the subject so that at least the obvious differences in context would be clear to you. > Yes, no kidding. There were references to terms and definitions which I > could at first glean fairly well, but as they built upon each other, the > references got buried upon each other, and I lost track. I have never > been a graduate mathematics student. Pardon me. I am not sure that works > to my disadvantage in this particular instance. My shackles are > non-mathematical. :) Forget about graduate level for now, is my very point. Please just learn the basics of the mathematical logic and set theory, which are the references that built up too fast for you in that chapter, so that you will understand that chapter and other graduate level mathematical logic (including non-standard analysis) as you move forward to study it. > This is why, when I choose a thread and participate, it takes forever to > end. I am looking in every direction, not to the front of the lecture > hall, to find the exotic critters under each rock and in each log. There > are underlying logical questions which you and others consider settled, I told you a long time ago that there is a tremendous amount of debate logical and mathematical-philosophical foundations. But you need to understand the rudiments of the subject so that you can understand what the different debators are saying. > Then perhaps your arrogance manifests itself in other ways. You might > want to think about that. I do. But if you ever catch me being an arrogant smart ass with people who know a lot more about this subject than I do and who have taken his time and effort to help me with explanations and formulas and proofs, then please let me know. > Welcome to the club, and while you're getting defensive about not being > the repository of all human or other knowledge, you might want to > consider, as I do while seeing othersget bent out of shape, that it > might be due to real world concerns, distractions, and needs. I am not judging you on how much time you don't put in. I'm judging you on how much you DO put in to post ignorant trash. Not even on the time you put in to post ignorant trash, but rather the ignorant trash itself. > shouldn't even be wasting my time on this, by any normal sensibilities, > given what's been and remains on my plate. Maybe that's true of you. Or > maybe you have no excuse at all. But I suspect you do. So, take you > time. But do check out Non-Standard Analysis, when you get the time. I'm > doing Boole and Robinson. We all choose our paths. This is not a matter of many paths. You can't properly do this without learing the rudiments. If you take a different path that skips the rudiments, then eventually you will have to come back to learn the rudiments anyway, and in the meanwhile you'll be burdened with a MISunderstanding of the advanced material, which is probably worse than no familiarity at all with it. Okay, probably there are a very few gifted people who can jump into the middle of advanced texts without having studied the basics, but I am not one of them, and I can tell from your POSTS that you are not one of them. MoeBlee === Subject: Re: An uncountable countable set > >> Eat me. Do you maintain that the two theories are compatible with each >> other? Is there, and also not, a smallest infinity. >They're not in conflict, becuase 'smallest infinite' means something > DIFFERENT in the different contexts. How many times will I say that > while you STILL refuse to hear it? >So, either smallest has two meanings, or infinite has tow meanings, or > both. Would you like to elucidate the matter by enumerating the various > definitions of small and infinite? A table might be nice... As many have said, infinite has many meanings. I'm afraid it isn't practical to produce a table. -- David Marcus === Subject: Re: An uncountable countable set >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. >You agreed it is not an ordering by cardinality. >Yes, and you asked me what I would think if it was derived from > transfinite set theory. You're tripping over your tail. When TO can explain how Robinson's ultrafilters can work without at least some uncountably infinite sets, TO will perhaps stop tripping over his own tail. > And the use of the > word 'infinite' is A DIFFERENT SENSE in the two different contexts. So > there is NO CONTRADICTION. >OHHHHH!!!!!!! WOW!!!!! NO CONTRADICTION!!!!! WITHIN MATHEMATICS!!! > THAT'S GREAT!!!!! WAS THAT THE PHONE????? No! That ringing in TO's ears is merely the reverberations through his empty head caused by his shouting. >Read Robinson for real and then comment. See how quickly you can absorb > his actual section on logical tools, and whether you might skip ahead > and then back. I have read it enough to know that TO has not understood whatever part of it he claims to have read. >You don't learn the > mathematical logic and set theory that are the basis for the material > and even pretty much ignore the mathemtatical logic and set theory that > the author himself summarizes in the book. (You need to start by > learning how to work in the predicate calculus), and (2) you don't > listen when someone tries to warn you about the confusions you are > making due to your not understanding the basis and context. Well, it doesn't help when they freak out because they can't handle > questioning certain iffy logical constructs. You're getting a tad edgy > lately, which I understand. I'm not floating on a cloud myself. TO is in his own cloud cuckooland, but whether he is floating or sinking is not apparent from within reality. >Why does the very mention of no smallest infinite elicit such bile, > then? If TO wants to avoid that bile he can merely stop claiming that there is no smallest infinite ordinal. > They're not in conflict, becuase 'smallest infinite' means something > DIFFERENT in the different contexts. How many times will I say that > while you STILL refuse to hear it? So, either smallest has two meanings, or infinite has tow meanings, or > both. Would you like to elucidate the matter by enumerating the various > definitions of small and infinite? A table might be nice... A head might be nice, too. Too bad TO doesn't have one. === Subject: Re: An uncountable countable set >> Uh, if Robinson's thesis is built upon transfinite set theory, then that >> is evidence right there that it's inconsistent, since you have a >> smallest infinity, omega, but Robinson has no smallest infinity. >We JUST agreed that 'smallest infinity' means two different things when > referring to ordinals and when referring to certain kinds of other > make sure this was clear, and then you agreeed, you NOW come back to > conflate the two ANYWAY! >Ahem. I said that Robinson's analysis seems to have nothing to do with > transfinitology. They appear to be unrelated. However, they cme to two > very different conclusions regarding a basic question: is there a > smallest infinite number? It seems clear to me there is not, for the > very same reason that Robinson uses: if there is an infinite number, you > can subtract 1 and get a different, smaller infinite number. It's the > same logic y'all use to argue that there's no largest finite. It's > correct. The Twilight Zone between finite and infinite CANNOT really be > pinpointed that way. >So, that's a verrry basic discrepancy. There is clearly a contradiction > between the two theories. They can't both be right about that, can they? > Is there, and at the same time is not, a smallest infinite number? >Tach me more about mathematical logic and consistency. We've been trying, but you don't seem to want to learn. Your question Is there a smallest infinite number? lacks context. You need to state what numbers you are considering. Lots of things can be constructed/defined that people refer to as numbers. However, these numbers differ in many details. If you assume that all subjects that use the word number are talking about the same thing, then it is hardly surprising that you would become confused. The two theories can both be right about the numbers that they are talking about, since they are talking about different things. To avoid confusion, the simplest solution is to be specific. We could say that there is a smallest infinite ordinal, but there is no smallest infinite non-standard real number. When phrased this way, no contradiction is apparent. The apparent contradiction is due to your using infinite number to mean both infinite ordinal and infinite non-standard real number. A similar problem would arise if you used the word cat to refer to both domestic cats and lions and were to say that Cats make good pets. -- David Marcus === Subject: Re: An uncountable countable set >> Uh, if Robinson's thesis is built upon transfinite set theory, then that >> is evidence right there that it's inconsistent, since you have a >> smallest infinity, omega, but Robinson has no smallest infinity. > We JUST agreed that 'smallest infinity' means two different things when > referring to ordinals and when referring to certain kinds of other > make sure this was clear, and then you agreeed, you NOW come back to > conflate the two ANYWAY! >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. They appear to be unrelated. However, they cme to two >> very different conclusions regarding a basic question: is there a >> smallest infinite number? It seems clear to me there is not, for the >> very same reason that Robinson uses: if there is an infinite number, you >> can subtract 1 and get a different, smaller infinite number. It's the >> same logic y'all use to argue that there's no largest finite. It's >> correct. The Twilight Zone between finite and infinite CANNOT really be >> pinpointed that way. >> So, that's a verrry basic discrepancy. There is clearly a contradiction >> between the two theories. They can't both be right about that, can they? > >> Is there, and at the same time is not, a smallest infinite number? >> Tach me more about mathematical logic and consistency. >We've been trying, but you don't seem to want to learn. I guess you missed the sarcastic tone there. I have a hard time taking lessons in logic from people who think things can happen in some kind of time without being anywhere in time. The contradiction here is glaring. >Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. I don't consider transfinite numbers to be real numbers at all. I'm not interested in that nonsense, to be honest. I see it as a dead end. If there is a definition for number in general, and for infinite, then there cannot both be a smallest infinite number and not be. >The two theories can both be right about the numbers that they are > talking about, since they are talking about different things. To avoid > confusion, the simplest solution is to be specific. We could say that > there is a smallest infinite ordinal, but there is no smallest infinite > non-standard real number. When phrased this way, no contradiction is > apparent. The apparent contradiction is due to your using infinite > number to mean both infinite ordinal and infinite non-standard real > number. >A similar problem would arise if you used the word cat to refer to > both domestic cats and lions and were to say that Cats make good pets. > claim to have a correct answer, and that any other interpretation is, as Virgil would say, WRONG!!!, well then, they should feel obligated to explain what's wrong with non-standard analysis, infinitesimals, infinite series, limits, infinite-case induction, and other approaches. If set theory is just a theory that's interesting in itself, fine, but the pompous attitude of set theorists towards anyone disagreeing with their nonsense only invites controversy. So, I don't really apologize for putting down transfinitology. What goes around comes around. :) TOny === Subject: Re: An uncountable countable set >> Tach me more about mathematical logic and consistency. >We've been trying, but you don't seem to want to learn. >I guess you missed the sarcastic tone there. TO missed the spell checker there, too. >I have a hard time taking > lessons in logic from people who think things can happen in some kind of > time without being anywhere in time. We have a hard time taking seriously someone who thinks he can stop time because he does not like what happens. Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >I don't consider transfinite numbers to be real numbers at all. I'm > not interested in that nonsense, to be honest. I see it as a dead end. Then why does TO deliberately conflate them with real numbers? >If there is a definition for number in general, and for infinite, > then there cannot both be a smallest infinite number and not be. As there is no definition for 'number' in general, but a variety of definitions in particular, one must particularize 'number' to a sort which allows infinites before specifying infinite'. TO does not do this. So he is then speaking nonsense. > claim to have a correct answer, and that any other interpretation is, > as Virgil would say, WRONG!!!, well then, they should feel obligated to > explain what's wrong with non-standard analysis, infinitesimals, > infinite series, limits, infinite-case induction, and other approaches. On the contrary. The standard stuff is already well established in texts and other works. If someone like TO wants to do anything non-standard, it is HE who must justify his case. It is the position of cranks that their claims stand without proofs unless they are refuted. It is the positions of non-cranks, that what has already been established and accepted shall persist until refuted. By those standards, TO is definitely cranky. === Subject: Re: An uncountable countable set > Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >I don't consider transfinite numbers to be real numbers at all. I'm > not interested in that nonsense, to be honest. I see it as a dead end. >If there is a definition for number in general, and for infinite, > then there cannot both be a smallest infinite number and not be. A moot point, since there is no definition for 'number' in general, as I just said. -- David Marcus === Subject: Re: An uncountable countable set >> Your question Is there a smallest infinite number? lacks context. You >> need to state what numbers you are considering. Lots of things can be >> constructed/defined that people refer to as numbers. However, these >> numbers differ in many details. If you assume that all subjects that >> use the word number are talking about the same thing, then it is >> hardly surprising that you would become confused. >> >> I don't consider transfinite numbers to be real numbers at all. I'm >> not interested in that nonsense, to be honest. I see it as a dead end. >> >> If there is a definition for number in general, and for infinite, >> then there cannot both be a smallest infinite number and not be. > A moot point, since there is no definition for 'number' in general, as > I just said. > -- > A very simple example is that there exists a smallest positive non-zero integer, but there does not exist a smallest positive non-zero real. If someone were to ask does there exist a smallest positive non-zero number?, the answer depends on what sort of numbers you are talking about. Stephen === Subject: Re: An uncountable countable set >> Your question Is there a smallest infinite number? lacks context. You >> need to state what numbers you are considering. Lots of things can be >> constructed/defined that people refer to as numbers. However, these >> numbers differ in many details. If you assume that all subjects that >> use the word number are talking about the same thing, then it is >> hardly surprising that you would become confused. > I don't consider transfinite numbers to be real numbers at all. I'm > not interested in that nonsense, to be honest. I see it as a dead end. >> If there is a definition for number in general, and for infinite, > then there cannot both be a smallest infinite number and not be. > >> A moot point, since there is no definition for 'number' in general, as >> I just said. > >> -- >A very simple example is that there exists a smallest positive > non-zero integer, but there does not exist a smallest positive > non-zero real. If someone were to ask does there exist a smallest > positive non-zero number?, the answer depends on what sort > of numbers you are talking about. >Stephen Like, perhaps, the Finlayson Numbers? :) === Subject: Re: An uncountable countable set > Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >> I don't consider transfinite numbers to be real numbers at all. I'm >> not interested in that nonsense, to be honest. I see it as a dead end. >> If there is a definition for number in general, and for infinite, >> then there cannot both be a smallest infinite number and not be. >>A moot point, since there is no definition for 'number' in general, as > I just said. >>-- > >> A very simple example is that there exists a smallest positive >> non-zero integer, but there does not exist a smallest positive >> non-zero real. If someone were to ask does there exist a smallest >> positive non-zero number?, the answer depends on what sort >> of numbers you are talking about. >> >> Stephen > Like, perhaps, the Finlayson Numbers? :) If they were sensibly defined then sure you could talk about them. Nothing Ross has ever said has made any sense to me, and I severely doubt there is any sense to it, but I could be wrong. The point is, there are different types of numbers, and statements that are true of one type of number need not be true of other types of numbers. Stephen === Subject: Re: An uncountable countable set >> Your question Is there a smallest infinite number? lacks context. You >> need to state what numbers you are considering. Lots of things can be >> constructed/defined that people refer to as numbers. However, these >> numbers differ in many details. If you assume that all subjects that >> use the word number are talking about the same thing, then it is >> hardly surprising that you would become confused. > I don't consider transfinite numbers to be real numbers at all. I'm > not interested in that nonsense, to be honest. I see it as a dead end. >> If there is a definition for number in general, and for infinite, > then there cannot both be a smallest infinite number and not be. > >> A moot point, since there is no definition for 'number' in general, as >> I just said. > >> -- >A very simple example is that there exists a smallest positive > non-zero integer, but there does not exist a smallest positive > non-zero real. If someone were to ask does there exist a smallest > positive non-zero number?, the answer depends on what sort > of numbers you are talking about. >Stephen >Like, perhaps, the Finlayson Numbers? :) Any set of numbers whose properties are known. Are the properties of Finlayson Numbers known to anyone except Ross himself? === Subject: Re: An uncountable countable set > Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >I don't consider transfinite numbers to be real numbers at all. I'm > not interested in that nonsense, to be honest. I see it as a dead end. >If there is a definition for number in general, and for infinite, > then there cannot both be a smallest infinite number and not be. > A moot point, since there is no definition for 'number' in general, as >> I just said. > -- >> A very simple example is that there exists a smallest positive >non-zero integer, but there does not exist a smallest positive >non-zero real. So non zero integers are not real? Or is this another zenmath conundrum? Just curious. > If someone were to ask does there exist a smallest >positive non-zero number?, the answer depends on what sort >of numbers you are talking about. Stephen ~v~~ === Subject: Re: An uncountable countable set >Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >> I don't consider transfinite numbers to be real numbers at all. I'm >> not interested in that nonsense, to be honest. I see it as a dead end. >> If there is a definition for number in general, and for infinite, >> then there cannot both be a smallest infinite number and not be. > A moot point, since there is no definition for 'number' in general, as > I just said. > -- > A very simple example is that there exists a smallest positive >> non-zero integer, but there does not exist a smallest positive >> non-zero real. So non zero integers are not real? Or is this another zenmath > conundrum? Just curious. > >> If someone were to ask does there exist a smallest >> positive non-zero number?, the answer depends on what sort >> of numbers you are talking about. >> Stephen >~v~~ The positive reals>0 include elements less than any positive natural>0, an infinite number in (0,1). === Subject: Re: An uncountable countable set >Your question Is there a smallest infinite number? lacks context. You > need to state what numbers you are considering. Lots of things can be > constructed/defined that people refer to as numbers. However, these > numbers differ in many details. If you assume that all subjects that > use the word number are talking about the same thing, then it is > hardly surprising that you would become confused. >> I don't consider transfinite numbers to be real numbers at all. I'm >> not interested in that nonsense, to be honest. I see it as a dead end. >> If there is a definition for number in general, and for infinite, >> then there cannot both be a smallest infinite number and not be. > A moot point, since there is no definition for 'number' in general, as > I just said. > -- > A very simple example is that there exists a smallest positive >> non-zero integer, but there does not exist a smallest positive >> non-zero real. So non zero integers are not real? Or is this another zenmath > conundrum? Just curious. > >> If someone were to ask does there exist a smallest >> positive non-zero number?, the answer depends on what sort >> of numbers you are talking about. >> Stephen >~v~~ >The positive reals>0 include elements less than any positive natural>0, > an infinite number in (0,1). Is this comment relevant in some way to what Stephen or I said or are you replying to Lester? -- David Marcus === Subject: Re: An uncountable countable set >A very simple example is that there exists a smallest positive >non-zero integer, but there does not exist a smallest positive >non-zero real. >So non zero integers are not real? That's a pretty impressive leap of illogic. > Or is this another zenmath conundrum? Just curious. -- David Marcus === Subject: Re: An uncountable countable set >>A very simple example is that there exists a smallest positive >>non-zero integer, but there does not exist a smallest positive >>non-zero real. >> >> So non zero integers are not real? > That's a pretty impressive leap of illogic. Using Lester IllLogic it is easy to prove that 2 is not prime. 2 is the largest even prime integer. There is no largest prime integer. Therefore 2 cannot be prime. Stephen === Subject: Re: An uncountable countable set <453d577b@news2.lightlink> <4540d449@news2.lightlink> <45417cb7@news2.lightlink> <45422534@news2.lightlink> A very simple example is that there exists a smallest positive >non-zero integer, but there does not exist a smallest positive >non-zero real. > So non zero integers are not real? That's a pretty impressive leap of illogic. Gosh, you obviously haven't seen Lester when he's in full swing. (Have _you_ searched sci.math for Zick transcendental?) Brian Chandler http://imaginatorium.org === Subject: Re: An uncountable countable set >A very simple example is that there exists a smallest positive >>non-zero integer, but there does not exist a smallest positive >>non-zero real. >>So non zero integers are not real? >> That's a pretty impressive leap of illogic. Gosh, you obviously haven't seen Lester when he's in full swing. (Have >_you_ searched sci.math for Zick transcendental?) No but obviously you have, Brian. ~v~~ === Subject: Re: An uncountable countable set >A very simple example is that there exists a smallest positive > non-zero integer, but there does not exist a smallest positive > non-zero real. >> So non zero integers are not real? > That's a pretty impressive leap of illogic. >> Gosh, you obviously haven't seen Lester when he's in full swing. (Have >> _you_ searched sci.math for Zick transcendental?) >No but obviously you have, Brian. >~v~~ I think Brian may have been in the swing at the time. Dunno. 01oo === Subject: Re: An uncountable countable set >>A very simple example is that there exists a smallest positive >non-zero integer, but there does not exist a smallest positive >non-zero real. > So non zero integers are not real? > That's a pretty impressive leap of illogic. >Gosh, you obviously haven't seen Lester when he's in full swing. (Have > _you_ searched sci.math for Zick transcendental?) I did briefly, but there are so many posts, I didn't read them all. -- David Marcus === Subject: Re: An uncountable countable set On Fri, 27 Oct 2006 16:23:44 -0400, >> >>A very simple example is that there exists a smallest positive >>non-zero integer, but there does not exist a smallest positive >>non-zero real. > So non zero integers are not real? >>That's a pretty impressive leap of illogic. >> >> Gosh, you obviously haven't seen Lester when he's in full swing. (Have >> _you_ searched sci.math for Zick transcendental?) I did briefly, but there are so many posts, I didn't read them all. You can read? ~v~~ === Subject: Re: An uncountable countable set >If the vase exists at noon, then it has an uncountable number of balls > labeled with infinite values. But, no infinite values are allowed i the > experiment, so this cannot happen, and noon is excluded. >> So did the North Koreans nuke the vase before noon? >> The only relevant issue is whether according to the rules set up in the >> problem, is each ball inserted before noon also removed before noon? >> An affirmative confirms that the vase is empty at noon. >> A negative directly violates the conditions of the problem. >> How does TO answer? > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. >> It is a good deal less inane and less nonsensical than trying to >> maintain, as TO and his ilk do, that a vase from which every ball has >> been removed before noon contains any balls at noon that have not been >> removed. > Ah, you are forgetting the balls labeled with infinite values. Those > balls haven't been removed before noon. Although, I must say I'm not too > clear on when they were added. > At noon >Where in the original problem does it say anything like that? It doesn't. It specifically excludes noon as a time in the experiment by specifying that all balls are finitely numbered and all events are finitely before noon. Duh. === Subject: Re: An uncountable countable set > >> If the vase exists at noon, then it has an uncountable number of > balls > labeled with infinite values. But, no infinite values are allowed i > the > experiment, so this cannot happen, and noon is excluded. >> So did the North Koreans nuke the vase before noon? >> The only relevant issue is whether according to the rules set up in >> the >> problem, is each ball inserted before noon also removed before noon? >> An affirmative confirms that the vase is empty at noon. >> A negative directly violates the conditions of the problem. >> How does TO answer? > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. >> It is a good deal less inane and less nonsensical than trying to >> maintain, as TO and his ilk do, that a vase from which every ball has >> been removed before noon contains any balls at noon that have not been >> removed. > Ah, you are forgetting the balls labeled with infinite values. Those > balls haven't been removed before noon. Although, I must say I'm not too > clear on when they were added. > At noon > Where in the original problem does it say anything like that? >> It doesn't. It specifically excludes noon as a time in the experiment by >> specifying that all balls are finitely numbered and all events are >> finitely before noon. Duh. >How can there be a before without what comes after? > How can there be a before noon without a noon? > Does God cause the sun to stand still and stop time? Uh, God doesn't limit Godself to finite numbers. === Subject: Re: An uncountable countable set >Also, supposing for the sake of argument that there are infinitely > number balls, if a ball is added at time -1/(2^floor(n/10)), and > removed > at time -1/(2^n)), then the balls added at time t=0, are those > where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > then -1/(2^n) = 0 (making some reasonable assumptions about how > arithmetic > on these infinite numbers works), so those balls are also removed at > noon and > never spend any time in the vase. >> Yes, the insertion/removal schedule instantly becomes infinitely fast >> in >> a truly uncountable way. The only way to get a handle on it is to >> explicitly state the level of infinity the iterations are allowed to >> achieve at noon. When the iterations are restricted to finite values, >> noon is never reached, but approached as a limit. > Suppose we only do an insertion or removal at t = 1/n for n a natural > number. What do you mean by noon is never reached? >> 1/n>0 > Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. >> But, I don't follow. Translating -1/n < 0 back into words, I get all > insertions and removals are before noon. However, I asked you what > noon is never reached means. Are you saying that noon is never > reached means that all insertions and removals are before noon? >> Yes, David. What else happens in this experiment besides insertions and >> removals of naturals at finite times before noon? If the infinite >> sequence of events is actually allowed to continue until t=0, then you >> are talking about events not indexed with natural numbers, so you're not >> talking about the same experiment. If noon is not allowed, and all times >> in the experiment are finitely before noon, well, at none of those times >> does the vase empty, as we all agree. This is why I am asking when this >> occurs. It can't, given the constraints of the problem. > Does your Yes at the beginning of your reply mean that you agree that > noon is never reached means that all insertions and removals are > before noon. By mean, I mean that that is what the words mean, not > that the two statements are equivalent or deducible from each other. > Yes, every event, every insertion or removal, happens at a specific time >> before noon. At each of those times, the vase is non-empty. Nothing else >> occurs, as far as insertions of removals. Is that clear enough? So, when >> does the vase become empty, and how? >The vase becomes empty in the usual way, by having everything in it > removed. And the time at which that finally has occurred is noon. Nothing is removed at noon. >The only relevant question is According to the rules set up by the > problem, is each ball inserted before noon also removed before noon? >An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the problem. >How does TO answer? That you are a broken record, and noon does not exist in the experiment. It is never the case that every ball inserted has been removed. === Subject: Re: An uncountable countable set > How is it, in your world, that when I specify times for all natural >> numbered >> balls, I am required to put in balls that don't have natural numbers? > The problem is that Tony thinks time is a function of the number of > insertions you've gone through. In order to get to any particular > time you have to perform the insertions up to that point. He then > thinks that if you want to get to noon, you have to have performed > some infinite (whatever that means) iterations, where balls without > natural numbers are inserted. That this is obviously not what the > problem statement says doesn't seem to bother him. Nor that it's > absolutely nothing like an intuitive picture of what time is. >> Time is ultimately irrelevant in this gedanken, but if it is to be >> considered, the constraints regarding time cannot be ignored. Events >> occurring in time must occupy at least one moment. How is time irrelevant when every action is specified by the time at > which it is to occur? Please specify the moment when the vase becomes empty. >The only relevant question is According to the rules set up in the > problem, is each ball inserted at a time before noon also removed at a > time before noon? >An affirmative answer confirms that the vase is empty at noon. Not if noon is proscribed the the problem itself, which it is. > A negative answer directly violates the conditions of the problem. >How does TO answer this question? >As usual, he avoids such relevant questions in his dogged pursuit of the > irrelevant. > Noon does not exist in the experiment, or else you have infinitely numbered balls. > Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. >> No, you have the concept of the independent variable bent. The number of >> balls is related to the time by a formula which works in both directions. >As time is a continuum and the numbers of balls in the vase is not, > there is no way of inverting the realtionship in the way that TO claims. Your times are as discontinuous as the number of balls, if no events can happen at any other moments than those specified. >> So, when does the vase become empty? Nothing can occur at noon, as far >> as ball removals. AT every time before noon, balls are in the vase. So, >> when does the vase become empty, and how? >The vase is empty when every ball has been removed, and that occurs at > noon. So, that occurs AT noon? The vase becomes empty, when no balls are being removed? Remember, every ball was removed BEFORE noon, and upon the removal of each and every ball, more balls resided still in the vase. So, how does the vase empty, when no balls are removed? > If you follow the sequence of insertions and removals you never get > to noon but this doesn't imply that noon is never reached, or that > iterations involving non-naturally numbered balls occur. It just > implies that all insertion and removal is performed before noon. >> Tony won't let himself understand this. He is delusional. His problem. > I won't let myself accept self-contradictory conclusions. > > At least not unless they are TO's own personal self-contradictory > conclusions. Like the existence of balls in a vase from which all balls > have been removed. > Like something occurring in time without at least a moment in which it occurred. >There is no >> moment at which this can occur. The problem is perfectly modeled by a >> divergent infinite series. No last ball can be removed without there >> having been an negative number of balls previously. You solution fails. >TO's assumption that there must be a last ball removed in order for all > balls to have been removed is part and parcel of his persistent delusion > that there must be a last (finite) natural number in order to have a set > of all (finite) natural numbers. No, that's what the problem implies when it claims to have completed the sequence of naturals. The fact is, with only naturally-numbered balls, one cannot have 1/n=0, and noon cannot be part of the experiment. === Subject: Re: An uncountable countable set > Noon does not exist in the experiment, or else you have infinitely > numbered balls. >> It is specifically mentioned in the experiment as the base time from >> which all actions are determined, so that if it does not exist then none >> of the actions can occur. >> If there is no noon then there can be no one minute before noon at which >> the first ball is inserted, so the vase is frozen in a state of >> emptiness. >That's a good point. Not particularly. Noon may be used as a time origin, but if all events happen such that n e N, and all events happen such that t=-1/n, then t<>0. Nothing can occur at noon, and the vase is not empty before noon, so it cannot be empty at noon. That's obvious. >Like something occurring in time without at least a moment in which it > occurred. >> In the physical world, nothing happens instantaneously. In the >> mathematical world, pretty much everything does. >> In the mathematical world of the experiment, the balls move in and out >> of the vase instantaneously, and must be allowed to do so or the >> experiment cannot be performed at all. >> So either things can happen instantaneously or the experiment impossible. >> If TO allows a finite change of number of balls in the vase to occur >> instantaneously, what is so difficult about allowing an infinite >> change in the number of balls to occur instantaneously? >That's another good point. > Yeah, sure, except that I never objected to an infinite number of events happening in one moment. I objected to them happening in NO moment. So, that's not a response. === Subject: Re: An uncountable countable set >Noon does not exist in the experiment, or else you have infinitely > numbered balls. >> It is specifically mentioned in the experiment as the base time from >> which all actions are determined, so that if it does not exist then none >> of the actions can occur. >> If there is no noon then there can be no one minute before noon at which >> the first ball is inserted, so the vase is frozen in a state of >> emptiness. >That's a good point. >Not particularly. Noon may be used as a time origin, but if all events > happen such that n e N, and all events happen such that t=-1/n, then > t<>0. Nothing can occur at noon, and the vase is not empty before noon, > so it cannot be empty at noon. That's obvious. WRONG! When every ball inserted before noon must be removed before noon, then to claim any balls still remain at noon is silly. But that's TO. Like something occurring in time without at least a moment in which it > occurred. >> In the physical world, nothing happens instantaneously. In the >> mathematical world, pretty much everything does. >> In the mathematical world of the experiment, the balls move in and out >> of the vase instantaneously, and must be allowed to do so or the >> experiment cannot be performed at all. >> So either things can happen instantaneously or the experiment impossible. >> If TO allows a finite change of number of balls in the vase to occur >> instantaneously, what is so difficult about allowing an infinite >> change in the number of balls to occur instantaneously? >That's another good point. Yeah, sure, except that I never objected to an infinite number of events > happening in one moment. I objected to them happening in NO moment. So, > that's not a response. === Subject: Re: An uncountable countable set >How is it, in your world, that when I specify times for all natural >> numbered >> balls, I am required to put in balls that don't have natural numbers? > The problem is that Tony thinks time is a function of the number of > insertions you've gone through. In order to get to any particular > time you have to perform the insertions up to that point. He then > thinks that if you want to get to noon, you have to have performed > some infinite (whatever that means) iterations, where balls without > natural numbers are inserted. That this is obviously not what the > problem statement says doesn't seem to bother him. Nor that it's > absolutely nothing like an intuitive picture of what time is. >> Time is ultimately irrelevant in this gedanken, but if it is to be >> considered, the constraints regarding time cannot be ignored. Events >> occurring in time must occupy at least one moment. >> How is time irrelevant when every action is specified by the time at > which it is to occur? >> Please specify the moment when the vase becomes empty. >It IS empty at noon, but not before. But I do not know what TO means by > becomes. Become: To assume a state not previously assumed. This can happen over a period, or in an instant, but it must happen sometime. If noon is the first moment when the vase is empty, then it emptied at noon, but nothing happens to the balls at noon. Contradiction. > The only relevant question is According to the rules set up in the > problem, is each ball inserted at a time before noon also removed at a > time before noon? >> An affirmative answer confirms that the vase is empty at noon. >> Not if noon is proscribed the the problem itself, which it is. >How so? I see nothing in the statement of the problem which proscribes > noon. Nothing can occur at noon because that implies 1/n=0, false for all natural numbers. > A negative answer directly violates the conditions of the problem. >> How does TO answer this question? >> As usual, he avoids such relevant questions in his dogged pursuit of the > irrelevant. > Noon does not exist in the experiment, or else you have infinitely >> numbered balls. >It is specifically mentioned in the experiment as the base time from > which all actions are determined, so that if it does not exist then none > of the actions can occur. No, time begins at -1, such that t(n)=-1/n. n never becomes infinite, so t never becomes 0. >If there is no noon then there can be no one minute before noon at which > the first ball is inserted, so the vase is frozen in a state of > emptiness. At t=-1=-1/n, n=1. Are you saying 1 is not a natural number? I thought the labels were the most important aspect of this for you. Now you want to ignore them? Huh! > Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. > >> No, you have the concept of the independent variable bent. The number of >> balls is related to the time by a formula which works in both directions. >Where does the problem say that the numbers on balls being moved > determines the time? Of each event? Where it says that ball n is inserted at time -1/n and removed at time -1/10n. That was a dumb question. > As time is a continuum and the numbers of balls in the vase is not, > there is no way of inverting the realtionship in the way that TO claims. >> Your times are as discontinuous as the number of balls, if no events can >> happen at any other moments than those specified. >That hardly means that there are no other times in between. >Time is a continuum. Or does TO claim that time is quantized? Where real time is continuous, there is always something happening. That's not the case here. The moments during events are a countable subset of the uncountable interval. >> So, when does the vase become empty? Nothing can occur at noon, as far >> as ball removals. AT every time before noon, balls are in the vase. So, >> when does the vase become empty, and how? > The vase is empty when every ball has been removed, and that occurs at > noon. >> So, that occurs AT noon? The vase becomes empty, when no balls are being >> removed? Remember, every ball was removed BEFORE noon, and upon the >> removal of each and every ball, more balls resided still in the vase. >> So, how does the vase empty, when no balls are removed? >So how is the vase be not empty after every ball is removed? There is no after. You are hiding your largest finite in a moment of infinite processing, but it's leaving a hole in your logic. If every natural is removed, then t=0, and infinite balls are added. That's how. > If you follow the sequence of insertions and removals you never get > to noon but this doesn't imply that noon is never reached, or that > iterations involving non-naturally numbered balls occur. It just > implies that all insertion and removal is performed before noon. >> Tony won't let himself understand this. He is delusional. His problem. > I won't let myself accept self-contradictory conclusions. > > At least not unless they are TO's own personal self-contradictory > conclusions. Like the existence of balls in a vase from which all balls > have been removed. >> Like something occurring in time without at least a moment in which it >> occurred. >In the physical world, nothing happens instantaneously. In the > mathematical world, pretty much everything does. Uh, yeah, at specific times. But there is no non-self-contradictory moment in the problem at which the emptying can occur. >In the mathematical world of the experiment, the balls move in and out > of the vase instantaneously, and must be allowed to do so or the > experiment cannot be performed at all. Of course, each event at a specific time. >So either things can happen instantaneously or the experiment impossible. So what? Happening instantaneously doesn't mean there is no moment in the event. It means there's only one. Without at least occupying one moment, an event does not happen in time. It's like trying to pretend you have a geometric figure that contains no points. The set of times that the vase becomes empty is null. >If TO allows a finite change of number of balls in the vase to occur > instantaneously, what is so difficult about allowing an infinite > change in the number of balls to occur instantaneously? >TO seems to swallow camels and strain at gnats. > There's nothing difficult about that. I agreed that, without removing balls, one gets an uncountable rate of increase at t=0. The problem is that there is no time when it can become empty in the original experiment, and there is no way it can become empty. Are you the camel or the gnat? > TO's assumption that there must be a last ball removed in order for all > balls to have been removed is part and parcel of his persistent delusion > that there must be a last (finite) natural number in order to have a set > of all (finite) natural numbers. >> No, that's what the problem implies when it claims to have completed the >> sequence of naturals. >The problem does not imply that to anyone except TO, so TO is still the > only one claiming a largest natural Uh, so, you HAVEN'T processed every natural in turn, by noon? What was your argument again? Sorry, I swatted it. === Subject: Re: An uncountable countable set >> Please specify the moment when the vase becomes empty. >It IS empty at noon, but not before. But I do not know what TO means by > becomes. >Become: To assume a state not previously assumed. Since the vase was empty to start with, it cannot later become empty after once having been empty, at least according to that definition. > The only relevant question is According to the rules set up in the > problem, is each ball inserted at a time before noon also removed at a > time before noon? >> An affirmative answer confirms that the vase is empty at noon. >> Not if noon is proscribed the the problem itself, which it is. >How so? I see nothing in the statement of the problem which proscribes > noon. >Nothing can occur at noon because that implies 1/n=0, false for all > natural numbers. Where in the gedankenexperiment is that required? >A negative answer directly violates the conditions of the problem. >> How does TO answer this question? >> As usual, he avoids such relevant questions in his dogged pursuit of the > irrelevant. > Noon does not exist in the experiment, or else you have infinitely >> numbered balls. Two assumptions both at variance with the original gedankenexperiment. >It is specifically mentioned in the experiment as the base time from > which all actions are determined, so that if it does not exist then none > of the actions can occur. >No, time begins at -1, such that t(n)=-1/n. n never becomes infinite, so > t never becomes 0. -1 is not a time unless there is a 0 from which to measure it. If there is no noon then there can be no one minute before noon at which > the first ball is inserted, so the vase is frozen in a state of > emptiness. >At t=-1=-1/n, n=1. Are you saying 1 is not a natural number? I am saying that there are no negative real numbers without a 0 from which to to mark them. > I thought > the labels were the most important aspect of this for you. Now you want > to ignore them? Huh! Non sequitur. That I want time to be properly measured does not mean require I want to ignore other things. >Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. > >> No, you have the concept of the independent variable bent. The number of >> balls is related to the time by a formula which works in both directions. >Where does the problem say that the numbers on balls being moved > determines the time? >Of each event? Where it says that ball n is inserted at time -1/n and > removed at time -1/10n. That was a dumb question. So the moving of the balls is determined by time, not the other way around. >As time is a continuum and the numbers of balls in the vase is not, > there is no way of inverting the realtionship in the way that TO claims. >> Your times are as discontinuous as the number of balls, if no events can >> happen at any other moments than those specified. >That hardly means that there are no other times in between. >Time is a continuum. Or does TO claim that time is quantized? >Where real time is continuous, there is always something happening. > That's not the case here. Sure there is, we are watching continuously! >The moments during events are a countable > subset of the uncountable interval. The moments of change are countable, but in between these moments the vase and the balls do not disappear, they still exist, they just don't move. > So how is the vase be not empty after every ball is removed? >There is no after. So TO wants to stop the clock? Is his face enough to do it? > You are hiding your largest finite in a moment of > infinite processing, but it's leaving a hole in your logic. TO may want a largest finite in his universe, in which clocks stop for no reason, but no one else need have one in theirs. >> Like something occurring in time without at least a moment in which it >> occurred. >In the physical world, nothing happens instantaneously. In the > mathematical world, pretty much everything does. >Uh, yeah, at specific times. But there is no non-self-contradictory > moment in the problem at which the emptying can occur. A bit of it happens each time a ball is removed, and it is completed when every ball has been removed, i.e., at noon. Or does TO's face stop the clock again? In the mathematical world of the experiment, the balls move in and out > of the vase instantaneously, and must be allowed to do so or the > experiment cannot be performed at all. >Of course, each event at a specific time. So either things can happen instantaneously or the experiment impossible. >So what? Happening instantaneously doesn't mean there is no moment in > the event. It means there's only one. Without at least occupying one > moment, an event does not happen in time. It's like trying to pretend > you have a geometric figure that contains no points. The set of times > that the vase becomes empty is null. But there is a noon, if there are any times at all, and at noon the vase holds no balls If TO allows a finite change of number of balls in the vase to occur > instantaneously, what is so difficult about allowing an infinite > change in the number of balls to occur instantaneously? >TO seems to swallow camels and strain at gnats. There's nothing difficult about that. I agreed that, without removing > balls, one gets an uncountable rate of increase at t=0. The problem is > that there is no time when it can become empty in the original > experiment, and there is no way it can become empty. Except by having all balls removed by noon. === Subject: Re: An uncountable countable set > Please specify the moment when the vase becomes empty. > It IS empty at noon, but not before. But I do not know what TO means by > becomes. >> Become: To assume a state not previously assumed. > > Since the vase was empty to start with, it cannot later become empty > after once having been empty, at least according to that definition. The only relevant question is According to the rules set up in the > problem, is each ball inserted at a time before noon also removed at a > time before noon? >> An affirmative answer confirms that the vase is empty at noon. >> Not if noon is proscribed the the problem itself, which it is. > How so? I see nothing in the statement of the problem which proscribes > noon. >> Nothing can occur at noon because that implies 1/n=0, false for all >> natural numbers. >Where in the gedankenexperiment is that required? > A negative answer directly violates the conditions of the problem. >> How does TO answer this question? >> As usual, he avoids such relevant questions in his dogged pursuit of the > irrelevant. > Noon does not exist in the experiment, or else you have infinitely >> numbered balls. >Two assumptions both at variance with the original gedankenexperiment. > It is specifically mentioned in the experiment as the base time from > which all actions are determined, so that if it does not exist then none > of the actions can occur. >> No, time begins at -1, such that t(n)=-1/n. n never becomes infinite, so >> t never becomes 0. >-1 is not a time unless there is a 0 from which to measure it. > If there is no noon then there can be no one minute before noon at which > the first ball is inserted, so the vase is frozen in a state of > emptiness. >> At t=-1=-1/n, n=1. Are you saying 1 is not a natural number? > I am saying that there are no negative real numbers without a 0 from > which to to mark them. > >> I thought >> the labels were the most important aspect of this for you. Now you want >> to ignore them? Huh! >Non sequitur. That I want time to be properly measured does not mean > require I want to ignore other things. > Obviously, time is an independent variable in this experiment and the > insertion or removal or location of balls is a function of time. That's > what the problem statement says: we have this thing called time which > is a real number and it goes from before noon to after noon and, at > certain specified times, things happen. There are only > naturally-numbered balls inserted and removed, always before noon. > Every ball is removed before noon. Therefore, the vase is empty. >> No, you have the concept of the independent variable bent. The number of >> balls is related to the time by a formula which works in both directions. > Where does the problem say that the numbers on balls being moved > determines the time? >> Of each event? Where it says that ball n is inserted at time -1/n and >> removed at time -1/10n. That was a dumb question. >So the moving of the balls is determined by time, not the other way > around. > As time is a continuum and the numbers of balls in the vase is not, > there is no way of inverting the realtionship in the way that TO claims. >> Your times are as discontinuous as the number of balls, if no events can >> happen at any other moments than those specified. > That hardly means that there are no other times in between. >> Time is a continuum. Or does TO claim that time is quantized? >> Where real time is continuous, there is always something happening. >> That's not the case here. >Sure there is, we are watching continuously! > >> The moments during events are a countable >> subset of the uncountable interval. The moments of change are countable, but in between these moments the > vase and the balls do not disappear, they still exist, they just don't > move. >So how is the vase be not empty after every ball is removed? >> There is no after. >So TO wants to stop the clock? Is his face enough to do it? > You are hiding your largest finite in a moment of >> infinite processing, but it's leaving a hole in your logic. >TO may want a largest finite in his universe, in which clocks stop for > no reason, but no one else need have one in theirs. >> Like something occurring in time without at least a moment in which it >> occurred. > In the physical world, nothing happens instantaneously. In the > mathematical world, pretty much everything does. >> Uh, yeah, at specific times. But there is no non-self-contradictory >> moment in the problem at which the emptying can occur. >A bit of it happens each time a ball is removed, and it is completed > when every ball has been removed, i.e., at noon. Or does TO's face stop the clock again? > In the mathematical world of the experiment, the balls move in and out > of the vase instantaneously, and must be allowed to do so or the > experiment cannot be performed at all. >> Of course, each event at a specific time. > So either things can happen instantaneously or the experiment impossible. >> So what? Happening instantaneously doesn't mean there is no moment in >> the event. It means there's only one. Without at least occupying one >> moment, an event does not happen in time. It's like trying to pretend >> you have a geometric figure that contains no points. The set of times >> that the vase becomes empty is null. >But there is a noon, if there are any times at all, and at noon the vase > holds no balls > If TO allows a finite change of number of balls in the vase to occur > instantaneously, what is so difficult about allowing an infinite > change in the number of balls to occur instantaneously? >> TO seems to swallow camels and strain at gnats. > There's nothing difficult about that. I agreed that, without removing >> balls, one gets an uncountable rate of increase at t=0. The problem is >> that there is no time when it can become empty in the original >> experiment, and there is no way it can become empty. >Except by having all balls removed by noon. I'm sorry, Virgil. This is just so full of nonsense I can't even find anything worth responding to. === Subject: Re: An uncountable countable set >TO seems to swallow camels and strain at gnats. > There's nothing difficult about that. I agreed that, without removing >> balls, one gets an uncountable rate of increase at t=0. The problem is >> that there is no time when it can become empty in the original >> experiment, and there is no way it can become empty. >Except by having all balls removed by noon. >I'm sorry, Virgil. This is just so full of nonsense I can't even find > anything worth responding to. The only relevant question is According to the rules set up in the gedankenexperiment, is each ball which is inserted into the vase before noon also removed from the vase before noon? An affirmative answer, as is required by the gedankenexperiment itself, confirms that the vase is empty at noon. === Subject: Re: An uncountable countable set > Does anything occur in the vase at noon? If not, then it should have the >> same state as before noon. > As which state before noon? > The state of non-emptiness that persists continually from t>=-1 until t<0. >What does TO mean by from t >= -1? > Does TO mean the same as from t = -1? > If so why not simply say so, and if not what does TO mean by it? And even more puzzling, what does TO mean by until t < 0? >Since t < 0 is true before the experiment starts, TO must mean from > the beginning of time. I mean during the interval [-1,0). Any less puzzled? Probably not. === Subject: Re: An uncountable countable set > As each ball n is removed, how many remain? >> 9n. > Can any be removed and leave an empty vase? >> Not sure what you are asking. > If, for all n e N, n>0, the number of balls remaining after n's removal > is 9n, does there exist any n e N which, after its removal, leaves 0? >> I don't know what you mean by after its removal? > Oh, I think this is clear, actually. Tony means: is there a ball (call > it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a > contradiction (following the stated rules of the experiment for the > moment) with the fact that ball P must have a pofnat p written on it, > and the pofnat 10p (or similar) must be inserted at the moment ball P > is removed. > >> I agree. If Tony means is there a ball P, removed at time t_P, such that >> the number of balls at time t_P is zero, then the answer is no. After >> all, I just agreed that the number of balls at the time when ball n is >> removed is 9n, and this is not zero for any n. >Now to you and me, this is all obvious, and no problem whatsoever, > because if ball P existed it would have to be the last natural > number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled > versions of Om mani padme hum, and protest that this is a Greatest > natural objection. For some reason he seems to accept that there is no > greatest natural number, yet feels that appealing to this fact in an > argument is somehow unfair. > >> The vase problem violates Tony's mental picture of a vase filling with >> water. If we are steadily adding more water than is draining out, how >> can all the water go poof at noon? Mental pictures are very useful, but >> sometimes you have to modify your mental picture to match the >> mathematics. Of course, when doing physics, we modify our mathematics to >> match the experiment, but the vase problem originates in mathematics >> land, so you should modify your mental picture to match the mathematics. >As someone else has pointed out, the balls and vase > are just an attempt to make this sound like a physical problem, > which it clearly is not, because you cannot physically move > an infinite number of balls in a finite time. It is just > a distraction. As you say, the problem originates in mathematics. > Any attempt to impose physical constraints on inherently unphysical > problem is just silly. >The problem could have been worded as follows: >Let IN = { n | -1/(2^floor(n/10) < 0 } > Let OUT = { n | -1/(2^n) } >What is | IN - OUT | ? >But that would not cause any fuss at all. >Stephen > It would still be inductively provable in my system that IN=OUT*10. === Subject: Re: An uncountable countable set >> As each ball n is removed, how many remain? > 9n. > Can any be removed and leave an empty vase? > Not sure what you are asking. >> If, for all n e N, n>0, the number of balls remaining after n's removal >> is 9n, does there exist any n e N which, after its removal, leaves 0? > I don't know what you mean by after its removal? >> Oh, I think this is clear, actually. Tony means: is there a ball (call >> it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a >> contradiction (following the stated rules of the experiment for the >> moment) with the fact that ball P must have a pofnat p written on it, >> and the pofnat 10p (or similar) must be inserted at the moment ball P >> is removed. > I agree. If Tony means is there a ball P, removed at time t_P, such that > the number of balls at time t_P is zero, then the answer is no. After > all, I just agreed that the number of balls at the time when ball n is > removed is 9n, and this is not zero for any n. >> Now to you and me, this is all obvious, and no problem whatsoever, >> because if ball P existed it would have to be the last natural >> number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled >> versions of Om mani padme hum, and protest that this is a Greatest >> natural objection. For some reason he seems to accept that there is no >> greatest natural number, yet feels that appealing to this fact in an >> argument is somehow unfair. > The vase problem violates Tony's mental picture of a vase filling with > water. If we are steadily adding more water than is draining out, how > can all the water go poof at noon? Mental pictures are very useful, but > sometimes you have to modify your mental picture to match the > mathematics. Of course, when doing physics, we modify our mathematics to > match the experiment, but the vase problem originates in mathematics > land, so you should modify your mental picture to match the mathematics. >> As someone else has pointed out, the balls and vase >> are just an attempt to make this sound like a physical problem, >> which it clearly is not, because you cannot physically move >> an infinite number of balls in a finite time. It is just >> a distraction. As you say, the problem originates in mathematics. >> Any attempt to impose physical constraints on inherently unphysical >> problem is just silly. >> The problem could have been worded as follows: >> Let IN = { n | -1/(2^floor(n/10) < 0 } >> Let OUT = { n | -1/(2^n) } >> What is | IN - OUT | ? >> But that would not cause any fuss at all. >> Stephen > It would still be inductively provable in my system that IN=OUT*10. >> So you actually think that there exists an integer n such that >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >> ? >> What might that integer be? >> Stephen >>How do you glean that from what I said? Your largest finite arguments > are very boring. >> >> How do I glean that? You claim that IN does not equal OUT. >> IN contains all n such that >> -1/(2^floor(n/10)) < 0 >> and OUT contains all n such that >> -1/(2^n) < 0 >> >> Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), >> so presumably IN is bigger than OUT, and IN contains elements >> that are not in OUT. The only way n can be an element of IN, >> but not of out is if >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 > Incorrect. For every n, finite or infinite, when the nth ball is > removed, 9n remain. Either you get to t=0, in which case all finite > balls are indeed gone, but replaced by uncountably many infinite balls, > or you don't get to noon. What are you talking about? I defined two sets. There are no balls or vases. There are simply the two sets IN = { n | -1/(2^floor(n/10)) < 0 } OUT = { n | -1/(2^n) < 0 } >> >> So apparently you do not think such an n exists, yet you >> think there are elements in IN that are not in OUT. >> Those are contradictory positions. > No, it is the only conclusion consistent with the notion that a proper > subset is alway smaller than the superset. It is contradictory to the > notion that a simple bijection indicates equal size for infinite sets. > The mapping formulas must be taken into account for proper comparison in > that case. But OUT is not a proper subset of IN, unless you believe that there exists an n such that -1/(2^floor(n/10)) < 0 but -1/(2^n) >= 0 If you claim that OUT is a proper subset of IN, you must be able to identify an element that is in IN that is not in OUT. Stephen === Subject: Re: An uncountable countable set > As each ball n is removed, how many remain? >> 9n. > Can any be removed and leave an empty vase? >> Not sure what you are asking. > If, for all n e N, n>0, the number of balls remaining after n's removal > is 9n, does there exist any n e N which, after its removal, leaves 0? >> I don't know what you mean by after its removal? > Oh, I think this is clear, actually. Tony means: is there a ball (call > it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a > contradiction (following the stated rules of the experiment for the > moment) with the fact that ball P must have a pofnat p written on it, > and the pofnat 10p (or similar) must be inserted at the moment ball P > is removed. >> I agree. If Tony means is there a ball P, removed at time t_P, such that >> the number of balls at time t_P is zero, then the answer is no. After >> all, I just agreed that the number of balls at the time when ball n is >> removed is 9n, and this is not zero for any n. > Now to you and me, this is all obvious, and no problem whatsoever, > because if ball P existed it would have to be the last natural > number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled > versions of Om mani padme hum, and protest that this is a Greatest > natural objection. For some reason he seems to accept that there is no > greatest natural number, yet feels that appealing to this fact in an > argument is somehow unfair. >> The vase problem violates Tony's mental picture of a vase filling with >> water. If we are steadily adding more water than is draining out, how >> can all the water go poof at noon? Mental pictures are very useful, but >> sometimes you have to modify your mental picture to match the >> mathematics. Of course, when doing physics, we modify our mathematics to >> match the experiment, but the vase problem originates in mathematics >> land, so you should modify your mental picture to match the mathematics. > As someone else has pointed out, the balls and vase > are just an attempt to make this sound like a physical problem, > which it clearly is not, because you cannot physically move > an infinite number of balls in a finite time. It is just > a distraction. As you say, the problem originates in mathematics. > Any attempt to impose physical constraints on inherently unphysical > problem is just silly. >> The problem could have been worded as follows: >> Let IN = { n | -1/(2^floor(n/10) < 0 } > Let OUT = { n | -1/(2^n) } >> What is | IN - OUT | ? >> But that would not cause any fuss at all. >> Stephen > It would still be inductively provable in my system that IN=OUT*10. > So you actually think that there exists an integer n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > ? >> What might that integer be? >> Stephen > How do you glean that from what I said? Your largest finite arguments >> are very boring. > How do I glean that? You claim that IN does not equal OUT. > IN contains all n such that > -1/(2^floor(n/10)) < 0 > and OUT contains all n such that > -1/(2^n) < 0 >> Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), > so presumably IN is bigger than OUT, and IN contains elements > that are not in OUT. The only way n can be an element of IN, > but not of out is if > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > >> Incorrect. For every n, finite or infinite, when the nth ball is >> removed, 9n remain. Either you get to t=0, in which case all finite >> balls are indeed gone, but replaced by uncountably many infinite balls, >> or you don't get to noon. >What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } > For each n e N, IN(n)=10*OUT(n). > So apparently you do not think such an n exists, yet you > think there are elements in IN that are not in OUT. > Those are contradictory positions. > >> No, it is the only conclusion consistent with the notion that a proper >> subset is alway smaller than the superset. It is contradictory to the >> notion that a simple bijection indicates equal size for infinite sets. >> The mapping formulas must be taken into account for proper comparison in >> that case. >But OUT is not a proper subset of IN, unless you believe that > there exists an n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 >If you claim that OUT is a proper subset of IN, you must > be able to identify an element that is in IN that is not in OUT. >Stephen For each n e N, IN(n)=10*OUT(n). You appear to think that the times are all that matters, but the times included in IN each apply to 10 elements, whereas the times in OUT each apply to only one. === Subject: Re: An uncountable countable set >What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } For each n e N, IN(n)=10*OUT(n). For each n in N, {x in IN: x <= n} >= {x in OUT: x <= n} >So apparently you do not think such an n exists, yet you > think there are elements in IN that are not in OUT. > Those are contradictory positions. > >> No, it is the only conclusion consistent with the notion that a proper >> subset is alway smaller than the superset. How is N a PROPER subset of N? > But OUT is not a proper subset of IN, unless you believe that > there exists an n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 >If you claim that OUT is a proper subset of IN, you must > be able to identify an element that is in IN that is not in OUT. >Stephen >For each n e N, IN(n)=10*OUT(n). For each n in N, {x in IN: x <= n} >= {x in OUT: x <= n} So that IN must be a subset of OUT. >You appear to think that the times are all that matters, but the times > included in IN each apply to 10 elements, whereas the times in OUT each > apply to only one. The only relevant question is According to the rules set up in the problem, is each ball inserted before noon also removed before noon? An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO find any balls in the vase after every ball has been removed? === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >For each n e N, IN(n)=10*OUT(n). Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT (n). So, you seem to be answering a question he didn't ask. Given Stephen's definitions of IN and OUT, is IN = OUT? -- David Marcus === Subject: Re: An uncountable countable set On Fri, 27 Oct 2006 12:01:19 -0400, >> What are you talking about? I defined two sets. There are no >> balls or vases. There are simply the two sets >> >> IN = { n | -1/(2^floor(n/10)) < 0 } >> OUT = { n | -1/(2^n) < 0 } >> >> For each n e N, IN(n)=10*OUT(n). Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT >(n). So, you seem to be answering a question he didn't ask. Given >Stephen's definitions of IN and OUT, is IN = OUT? According to MoeBlee's recent lectures on the subject of exhaustive mathematical definitions one cannot simply define IN and OUT, one must use a placeholder such as IN(x) and OUT(x) to establish the domain of discourse. Not exactly to my taste but there it is. And I'm sure Moe must be right because he says he is. ~v~~ === Subject: Re: An uncountable countable set <4540d016@news2.lightlink> <4540fa4b@news2.lightlink> <454184bc@news2.lightlink> <454227d7@news2.lightlink> mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled interpretations as if they are what I said. MoeBlee === Subject: Re: An uncountable countable set >> According to MoeBlee's recent lectures on the subject of exhaustive >> mathematical definitions one cannot simply define IN and OUT, one must >> use a placeholder such as IN(x) and OUT(x) to establish the domain of >> discourse. Please stop mangling what I've said and then representing your mangled >interpretations as if they are what I said. I will if you'll just stop doubletalking, Moe. ~v~~ === Subject: Re: An uncountable countable set <4540d016@news2.lightlink> <4540fa4b@news2.lightlink> <454184bc@news2.lightlink> <454227d7@news2.lightlink> Please stop mangling what I've said and then representing your mangled >interpretations as if they are what I said. I will if you'll just stop doubletalking, Moe. No, you'll just keep doing it until, like a child pestering to play peek-a-boo, constantly tugging on the coats of adults, you tire of your own silly game. MoeBlee === Subject: Re: An uncountable countable set <4540d016@news2.lightlink> <4540fa4b@news2.lightlink> <454184bc@news2.lightlink> <454227d7@news2.lightlink> mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled versions as if they are what I said. MoeBlee === Subject: Re: An uncountable countable set <4540d016@news2.lightlink> <4540fa4b@news2.lightlink> <454184bc@news2.lightlink> <454227d7@news2.lightlink> mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled versions as if they are what I said. (And your claim that that is a perfectly acceptable forensic modality is typical Zickian rot.) Or, you'll just keep doing it until, like a child demanding to play peek-a-boo and forever tugging on the coats of adults, you tire of your own silly game. MoeBlee === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). >Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? > Yes, all elements are the same n, which are finite n. There is a simple bijection. But, as in all infinite bijections, the formulaic relationship between the sets is lost. === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). >Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? Yes, all elements are the same n, which are finite n. There is a simple > bijection. But, as in all infinite bijections, the formulaic > relationship between the sets is lost. What never existed cannor be lost. === Subject: Re: An uncountable countable set >What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). > Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? > Yes, all elements are the same n, which are finite n. There is a simple >> bijection. But, as in all infinite bijections, the formulaic >> relationship between the sets is lost. >What never existed cannor be lost. Well, it's good to know that there is no mapping from the naturals to the evens by the formulaic relation f(x)=2x. That clears up a lot of problems.... === Subject: Re: An uncountable countable set > Well, it's good to know that there is no mapping from the naturals to > the evens by the formulaic relation f(x)=2x. That clears up a lot of > problems.... TO again assumes things not said. === Subject: Re: An uncountable countable set >> What are you talking about? I defined two sets. There are no >> balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } >> OUT = { n | -1/(2^n) < 0 } > For each n e N, IN(n)=10*OUT(n). >> >> Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT >> (n). So, you seem to be answering a question he didn't ask. Given >> Stephen's definitions of IN and OUT, is IN = OUT? >>Yes, all elements are the same n, which are finite n. There is a simple > bijection. But, as in all infinite bijections, the formulaic > relationship between the sets is lost. What formulaic relationship? There are two sets. The members of each set are identified by a predicate. If an element satifies the predicate, it is in the set. If it does not, it is not in the set. I could define different sets with different predicates. For example, A = { n | 1+n > 0 } B = { n | 2*n >= n } C = { n | sin(n*pi)=0 } Are these sets formulaically related? Assuming that n is restricted to non-negative integers, does A differ from B, C, IN, or OUT? Stephen === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). >Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? >Yes, all elements are the same n, which are finite n. There is a simple > bijection. But, as in all infinite bijections, the formulaic > relationship between the sets is lost. Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? (The vertical lines denote cardinality.) -- David Marcus === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). > Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? >> Yes, all elements are the same n, which are finite n. There is a simple >> bijection. But, as in all infinite bijections, the formulaic >> relationship between the sets is lost. >Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > (The vertical lines denote cardinality.) > Um, before I answer that question, I think you need to define what you mean by |IN - OUT| =0. How are you measuring IN and OUT, and how do you define '-' on these numbers? === Subject: Re: An uncountable countable set > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets >> IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } >> For each n e N, IN(n)=10*OUT(n). > Stephen defined sets IN and OUT. He didn't define sets IN(n) and OUT > (n). So, you seem to be answering a question he didn't ask. Given > Stephen's definitions of IN and OUT, is IN = OUT? >> Yes, all elements are the same n, which are finite n. There is a simple >> bijection. But, as in all infinite bijections, the formulaic >> relationship between the sets is lost. >Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > (The vertical lines denote cardinality.) >Um, before I answer that question, I think you need to define what you > mean by |IN - OUT| =0. How are you measuring IN and OUT, and how do > you define '-' on these numbers? IN and OUT are sets, not numbers. For any two sets A and B, the difference, denoted by A - B, is defined to be the set of elements in A that are not in B. Formally, A - B := {x| x in A and x not in B} Note that the difference of two sets is again a set. For any set, the notation |A| means the cardinality of A. So, saying that |A| = 0 is equivalent to saying that A is the empty set. In particular, for any set A, we have |A - A| = 0. -- David Marcus === Subject: Re: An uncountable countable set > Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > (The vertical lines denote cardinality.) Um, before I answer that question, I think you need to define what you > mean by |IN - OUT| =0. How are you measuring IN and OUT, and how do > you define '-' on these numbers? I suspect that David means by IN - OUT the set difference defined as {x in IN: not x in OUT} In this NG, this set difference is sometimes indicated with the backslash, as IN OUT. === Subject: Re: An uncountable countable set >Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? >(The vertical lines denote cardinality.) >Um, before I answer that question, I think you need to define what you > mean by |IN - OUT| =0. How are you measuring IN and OUT, and how do > you define '-' on these numbers? >I suspect that David means by IN - OUT the set difference defined as > {x in IN: not x in OUT} >In this NG, this set difference is sometimes indicated with the > backslash, as IN OUT. I sometimes use that notation, too! -- David Marcus === Subject: Re: An uncountable countable set >> As each ball n is removed, how many remain? > 9n. > Can any be removed and leave an empty vase? > Not sure what you are asking. >> If, for all n e N, n>0, the number of balls remaining after n's >> removal >> is 9n, does there exist any n e N which, after its removal, leaves >> 0? > I don't know what you mean by after its removal? >> Oh, I think this is clear, actually. Tony means: is there a ball >> (call >> it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a >> contradiction (following the stated rules of the experiment for the >> moment) with the fact that ball P must have a pofnat p written on >> it, >> and the pofnat 10p (or similar) must be inserted at the moment ball >> P >> is removed. > I agree. If Tony means is there a ball P, removed at time t_P, such > that > the number of balls at time t_P is zero, then the answer is no. After > all, I just agreed that the number of balls at the time when ball n > is > removed is 9n, and this is not zero for any n. >> Now to you and me, this is all obvious, and no problem whatsoever, >> because if ball P existed it would have to be the last natural >> number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled >> versions of Om mani padme hum, and protest that this is a Greatest >> natural objection. For some reason he seems to accept that there is >> no >> greatest natural number, yet feels that appealing to this fact in an >> argument is somehow unfair. > The vase problem violates Tony's mental picture of a vase filling > with > water. If we are steadily adding more water than is draining out, how > can all the water go poof at noon? Mental pictures are very useful, > but > sometimes you have to modify your mental picture to match the > mathematics. Of course, when doing physics, we modify our mathematics > to > match the experiment, but the vase problem originates in mathematics > land, so you should modify your mental picture to match the > mathematics. >> As someone else has pointed out, the balls and vase >> are just an attempt to make this sound like a physical problem, >> which it clearly is not, because you cannot physically move >> an infinite number of balls in a finite time. It is just >> a distraction. As you say, the problem originates in mathematics. >> Any attempt to impose physical constraints on inherently unphysical >> problem is just silly. >> The problem could have been worded as follows: >> Let IN = { n | -1/(2^floor(n/10) < 0 } >> Let OUT = { n | -1/(2^n) } >> What is | IN - OUT | ? >> But that would not cause any fuss at all. >> Stephen > It would still be inductively provable in my system that IN=OUT*10. >> So you actually think that there exists an integer n such that >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >> ? >> What might that integer be? >> Stephen >>How do you glean that from what I said? Your largest finite arguments > are very boring. >> >> How do I glean that? You claim that IN does not equal OUT. >> IN contains all n such that >> -1/(2^floor(n/10)) < 0 >> and OUT contains all n such that >> -1/(2^n) < 0 >> >> Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), >> so presumably IN is bigger than OUT, and IN contains elements >> that are not in OUT. The only way n can be an element of IN, >> but not of out is if >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >Incorrect. For every n, finite or infinite, when the nth ball is > removed, 9n remain. Either you get to t=0, in which case all finite > balls are indeed gone, but replaced by uncountably many infinite balls, > or you don't get to noon. >What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } > >> >> So apparently you do not think such an n exists, yet you >> think there are elements in IN that are not in OUT. >> Those are contradictory positions. >No, it is the only conclusion consistent with the notion that a proper > subset is alway smaller than the superset. It is contradictory to the > notion that a simple bijection indicates equal size for infinite sets. > The mapping formulas must be taken into account for proper comparison in > that case. >But OUT is not a proper subset of IN, unless you believe that > there exists an n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 >If you claim that OUT is a proper subset of IN, you must > be able to identify an element that is in IN that is not in OUT. >Stephen That reminds me of the story about the twin baby skunks. === Subject: Re: An uncountable countable set >> As someone else has pointed out, the balls and vase >> are just an attempt to make this sound like a physical problem, >> which it clearly is not, because you cannot physically move >> an infinite number of balls in a finite time. It is just >> a distraction. As you say, the problem originates in mathematics. >> Any attempt to impose physical constraints on inherently unphysical >> problem is just silly. >> The problem could have been worded as follows: >> Let IN = { n | -1/(2^floor(n/10) < 0 } >> Let OUT = { n | -1/(2^n) } >> What is | IN - OUT | ? >> But that would not cause any fuss at all. >> Stephen > It would still be inductively provable in my system that IN=OUT*10. >> So you actually think that there exists an integer n such that >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >> ? >> What might that integer be? >> Stephen >>How do you glean that from what I said? Your largest finite arguments > are very boring. >> >> How do I glean that? You claim that IN does not equal OUT. >> IN contains all n such that >> -1/(2^floor(n/10)) < 0 >> and OUT contains all n such that >> -1/(2^n) < 0 >> >> Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), >> so presumably IN is bigger than OUT, and IN contains elements >> that are not in OUT. The only way n can be an element of IN, >> but not of out is if >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >Incorrect. For every n, finite or infinite, when the nth ball is > removed, 9n remain. Either you get to t=0, in which case all finite > balls are indeed gone, but replaced by uncountably many infinite balls, > or you don't get to noon. >What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } It is interesting that when we try to ask Tony a question that doesn't mention balls or vases or time, his answer involves balls, vases, and time. I'm afraid to ask what 1 + 1 is because the answer might be noon doesn't exist. -- David Marcus === Subject: Re: An uncountable countable set On Fri, 27 Oct 2006 01:37:04 -0400, [. . .] >It is interesting that when we try to ask Tony a question that doesn't >mention balls or vases or time, his answer involves balls, vases, and >time. I'm afraid to ask what 1 + 1 is because the answer might be noon >doesn't exist. So if the definition for 1+1 entails 1(x)+1(x) balls don't lie in the domain of discourse for 1+1? Curious to say the least. ~v~~ === Subject: Re: An uncountable countable set > On Fri, 27 Oct 2006 01:37:04 -0400, >[. . .] > >> It is interesting that when we try to ask Tony a question that doesn't >> mention balls or vases or time, his answer involves balls, vases, and >> time. I'm afraid to ask what 1 + 1 is because the answer might be noon >> doesn't exist. >So if the definition for 1+1 entails 1(x)+1(x) balls don't lie > in the domain of discourse for 1+1? Curious to say the least. >~v~~ 1+1=2, of course, and noon doesn't exist. I think that's what David wants to hear. It makes him feel good. :) Tony === Subject: Re: An uncountable countable set >> As each ball n is removed, how many remain? > 9n. > Can any be removed and leave an empty vase? > Not sure what you are asking. >> If, for all n e N, n>0, the number of balls remaining after n's removal >> is 9n, does there exist any n e N which, after its removal, leaves 0? > I don't know what you mean by after its removal? >> Oh, I think this is clear, actually. Tony means: is there a ball (call >> it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a >> contradiction (following the stated rules of the experiment for the >> moment) with the fact that ball P must have a pofnat p written on it, >> and the pofnat 10p (or similar) must be inserted at the moment ball P >> is removed. >I agree. If Tony means is there a ball P, removed at time t_P, such that > the number of balls at time t_P is zero, then the answer is no. After > all, I just agreed that the number of balls at the time when ball n is > removed is 9n, and this is not zero for any n. > >> Now to you and me, this is all obvious, and no problem whatsoever, >> because if ball P existed it would have to be the last natural >> number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled >> versions of Om mani padme hum, and protest that this is a Greatest >> natural objection. For some reason he seems to accept that there is no >> greatest natural number, yet feels that appealing to this fact in an >> argument is somehow unfair. >The vase problem violates Tony's mental picture of a vase filling with > water. If we are steadily adding more water than is draining out, how > can all the water go poof at noon? Mental pictures are very useful, but > sometimes you have to modify your mental picture to match the > mathematics. Of course, when doing physics, we modify our mathematics to > match the experiment, but the vase problem originates in mathematics > land, so you should modify your mental picture to match the mathematics. > I disagree. When you formulate a theory, whether scientific or mathematical, the goal should be to draw conclusions in line with observations. In science, it's no problem to disprove a theory, if there is a verifiable situation which it predicts incorrectly. When it comes to math, there is no such test, but the whole of mathematics should be consistent, and where one theory contradicts another, that's an indication that one or the other is less than correct. In the case of questions regarding oo, no theory should cause blatant contradictions, such as an event occurring but there being no moment in time during which it is occurring. If you have to accept such a conclusion to salvage a theory, it's time to look for alternatives that don't require you to sacrifice common sense and basic logic. This is just one example of where this theory goes wrong, along with proper subsets of the same size as the superset, and the concept of a smallest infinity. I simply don't accept the theory, because its conclusions are bizarre. === Subject: Re: An uncountable countable set > >> As each ball n is removed, how many remain? > 9n. > Can any be removed and leave an empty vase? > Not sure what you are asking. >> If, for all n e N, n>0, the number of balls remaining after n's removal >> is 9n, does there exist any n e N which, after its removal, leaves 0? > I don't know what you mean by after its removal? >> Oh, I think this is clear, actually. Tony means: is there a ball (call >> it ball P) such that after the removal of ball P, zero balls remain. >> The answer is No, obviously. If there were, it would be a >> contradiction (following the stated rules of the experiment for the >> moment) with the fact that ball P must have a pofnat p written on it, >> and the pofnat 10p (or similar) must be inserted at the moment ball P >> is removed. > I agree. If Tony means is there a ball P, removed at time t_P, such that > the number of balls at time t_P is zero, then the answer is no. After > all, I just agreed that the number of balls at the time when ball n is > removed is 9n, and this is not zero for any n. > Now to you and me, this is all obvious, and no problem whatsoever, >> because if ball P existed it would have to be the last natural >> number, and there is no last natural number. >> Tony has a strange problem with this, causing him to write mangled >> versions of Om mani padme hum, and protest that this is a Greatest >> natural objection. For some reason he seems to accept that there is no >> greatest natural number, yet feels that appealing to this fact in an >> argument is somehow unfair. > The vase problem violates Tony's mental picture of a vase filling with > water. If we are steadily adding more water than is draining out, how > can all the water go poof at noon? Mental pictures are very useful, but > sometimes you have to modify your mental picture to match the > mathematics. Of course, when doing physics, we modify our mathematics to > match the experiment, but the vase problem originates in mathematics > land, so you should modify your mental picture to match the mathematics. > I disagree. When you formulate a theory, whether scientific or >> mathematical, the goal should be to draw conclusions in line with >> observations. In science, it's no problem to disprove a theory, if there >> is a verifiable situation which it predicts incorrectly. When it comes >> to math, there is no such test, but the whole of mathematics should be >> consistent, and where one theory contradicts another, that's an >> indication that one or the other is less than correct. >That depends. >If the apparently contradictory results follow from different axiom > systems, they may both be quite valid. They cannot be mutually consistent, and a larger mathematical system > including both cannot be internally consistent. Why not? Since each is based on assumptions which are not required to be true, different assumptions may lead to different conclusions without any problems. > The universe is > consistent, and math creates and describes it. I'll take a side of > logic, and hold the contradictions. But despite his high moral tone above, TO always takes the side opposed to logic and includes his own contradictions. === Subject: Intuitive Set Theory Hi All, This is a set theory that I have constructed. I called it Intuitive Set TheoryI.. It simply defines sets as containers and member in a set as what fulfill the requirement for inclusion to the set. Set identity is dependent on the members it contains. This set theory is a simple set theory, and it is intended to be the Primordial set from which all other sets emerge. So sets that belong to this set theory that fulfill ZFC are called ZFC sets, those that fulfill NBG are called NBG sets, in general any set fulfilling X set theory are called X sets. All of them are in I. Have fun at: http://zaljohar.tripod/I.doc Zuhair === Subject: Intuitive Set Theory Hi All, This is a set theory that I have constructed. I called it Intuitive Set TheoryI.. It simply defines sets as containers and member in a set as what fulfill the requirement for inclusion to the set. Set identity is dependent on the members it contains. This set theory is a simple set theory, and it is intended to be the Primordial set from which all other sets emerge. So sets that belong to this set theory that fulfill ZFC are called ZFC sets, those that fulfill NBG are called NBG sets, in general any sets fulfilling X set theory are called X sets. All of them are in I. Have fun at: http://zaljohar.tripod/I.doc Zuhair === Subject: Re: representing ideal as an intersection of prime ideals <4533C062.7010907@web.de> <4533C539.5030804@web.de> <4533CF95.4000306@web.de> <4537EFD9.4080305@web.de >> I have the following question: >> Let I be an ideal I = (xz - y^2, z^3 - x^5). I need to represent I as >> an intersection of prime ideals which are ideal quoitients of I and >> this would prove that I is radical. > I assume that you are working in the polynomial ring A[x,y,z] with a > commutative ring with unity. Most likely A is a field. Is it correct > that you want to represent I as an intersection of prime ideals > containing I or, equivalently, to show that A[x,y,z]/I is reduced? >> Yes, A = C - complex field. >> What do you mean by reduced ideal? > A ring is called reduced iff zero is the only nilpotent. Then: > A[x,y,z]/I is reduced iff I is radical, i.e. I = rad(I). >> Well, to show that R=k[x,y,z]/I (with a field k) is reduced note that >> R'=k[x,z]/(z^3-x^5) is an integral domain and R is a free R'-module with >> basis 1 and y. >> Now take an element f in R with f^n=0 (for some n>0). Then expand f^n >> wrt {1,y} and analyze the y-coefficient of f^n in R'. ... ... Hence f=0 >> (in R). > At first i thought that i know how to proceed with you idea here, but > now realized that probably i don't. > So, if f(x,y,z) = g_1(x,z) + g_2(x,z)y with g_1,g_2 in R', then > if f^n = 0 then there exists g,h in k[x,y,z] such that > f^n = f(x,y,z)*(xz-y^2) + g(x,y,z)*(z^3-x^5). Typo: f^n = h(x,y,z)*(xz-y^2) + g(x,y,z)*(z^3-x^5) Nevertheless it is not needed to calculate in k[x,y,z]. You can > calculate in *R* and *R'* which is a free module over R' wrt the basis > {1,y}. > the coefficient in y of f^n is n*g_1^(n-1) (x,z) * g_2(x,z) * y ... and this implies that n*g_1^(n-1)*g_2 = 0 in R'. Now use that R' is > an integral domain and finally conclude that f = 0. > and if > we consider it in R' we may > drop the multiplier g(x,y,z)*(z^3-x^5) because everything in (z^3 - > x^5) is zero in R', so there > must be a summand of f(x,y,z) : k(x,z)*y and then > n*g_1^(n-1) (x,z) * g_2(x,z) = k(x,z) *xz. I am not sure how you derived this, but note that you can calculate in > the quotient rings R and R' having certain properties. Hi again, i'm sorry for reviving old threads. But is think that there still must be something wrong with you idea... at first suppose (g_1 + g_2y)^n = 0 (*) . where g_1,g_2 in R' and R is considered as a free module over R' with respect to the basis {1,y} as you said. So, exapnding the expression we know that in R, y^2 = xz, so that the only powers in (*) which contain y are y,y^3,y^5,... and last time i complicated, namely we have : y*( ng_1^(n-1) * g_2 + C_n^3 g_1^(n-3)g_2^2 * xz + ... +) = 0, and in fact at least for me it is not evident that it follows that g_1 = g_2 = 0. Is it the idea you actually meant? or maybe i did somethign wrong. > And i don't know how to argue from this that at least one of the g_1 or > g_2 is zero in R'. > Maybe i did something wrong. > I'd be grateful fi you could help me to figure out. > >> J. >For your original question on fractional ideals this link might be > helpful: http://www.imsc.res.in/~kapil/geometry/caag/primary.html. >HTH. >J. === Subject: Re: Cantor Confusion certainly said things very similar. Below are a few messages that I > found. There are probably others. In the first, he says that standard mathematics contains a > contradiction. In the next two, he states there are internal > contradictions of set theory. In the next, I say that he says that > standard mathematics contains a contradiction, and he does not > dispute this. argument to see whether it does sustain his claim about set theory, as I mentioned specifically Z set theory. How rude. He tells you he's going to demonstrate something IN set theory, so, ON THAT BASIS, you take the time to ponder his argument, then he just pulls the rug out from under by saying that it's something outside of ZFC but that covers ZFC. MoeBlee === Subject: Re: Cantor Confusion > Not sure if he ever said precisely within Z set theory, but he > certainly said things very similar. Below are a few messages that I > found. There are probably others. > In the first, he says that standard mathematics contains a > contradiction. In the next two, he states there are internal > contradictions of set theory. In the next, I say that he says that > standard mathematics contains a contradiction, and he does not > dispute this. >argument to see whether it does sustain his claim about set theory, as > I mentioned specifically Z set theory. >How rude. He tells you he's going to demonstrate something IN set > theory, so, ON THAT BASIS, you take the time to ponder his argument, > then he just pulls the rug out from under by saying that it's something > outside of ZFC but that covers ZFC. I seriously doubt he understands the difference. He doesn't seem to really understand that modern mathematics rests on an axiomatic foundation. And, that there are certain agreed upon rules of argument (codified in the axioms) that people use. If someone wants to use some other rule of argument, they should clearly state that they are doing so. This is just a common sense prerequisite for communication. -- David Marcus === Subject: Re: Cantor Confusion [. . .] >You need transfinity when you want to show that something that holds in >the finite case also is valid in the infinite case. Induction will not >show that 0.111... is rational, it can only show that all the finite >initial parts are rational. And I again note that the notation 0.111... >(in the decimals) has only meaning due to the definition of that notation. However one can certainly show the square root of 2 without transfinity through rac construction even though its decimal expansion is infinite. ~v~~ === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl >You need transfinity when you want to show that something that holds in >the finite case also is valid in the infinite case. Induction will not >show that 0.111... is rational, it can only show that all the finite >initial parts are rational. And I again note that the notation 0.111... >(in the decimals) has only meaning due to the definition of that notation. > > However one can certainly show the square root of 2 without > transfinity through rac construction even though its decimal expansion > is infinite. You need not tell that to me. You should tell that to Wolfgang Mueckenheim who insists that sqrt(2) does not exist because it is impossible to know all the decimals in its decimal expansion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion Within the *real* numbers the limit does exist. And a decimal number is >nothing more nor less than a representative of an equivalence classes. >So we are agian at this point: The real numbers do exist. For the real > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > is no application of Cantor's argument. And again we are back at this point. You do not comprehend what I am >writing. The reason real numbers exist is that there are sequences of >rationals that come arbitrarily close to each other. Technically the reason real numbers exist is that there are straight line and curve segments. It has nothing to do with decimal expansions. > The reason that >a decimal expansion is a representative of a real number is because the >sequence of finitely terminations of that number is a sequence of >rationals that comes arbitrarily close to other such sequences and so >falls in an equivalence class. *No* limit is involved in all of this. What is the sequence of finitely terminations (sic) for transcendentals? Are you suggesting the sequence itself is finite? ~v~~ === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > > >Within the *real* numbers the limit does exist. And a decimal number is >nothing more nor less than a representative of an equivalence classes. > >So we are agian at this point: The real numbers do exist. For the real >numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere >is no application of Cantor's argument. > >And again we are back at this point. You do not comprehend what I am >writing. The reason real numbers exist is that there are sequences of >rationals that come arbitrarily close to each other. > > Technically the reason real numbers exist is that there are straight > line and curve segments. It has nothing to do with decimal expansions. I do not agree with the first, but I agree with the second. It is Wolfgang Mueckenheim who thinks that the reals have everything to do with decimal expansion. For the first, I can state that you do not need curves for sqrt(2), and that I have no idea how to use curves to define 'e'. > The reason that >a decimal expansion is a representative of a real number is because the >sequence of finitely terminations of that number is a sequence of >rationals that comes arbitrarily close to other such sequences and so >falls in an equivalence class. *No* limit is involved in all of this. > > What is the sequence of finitely terminations (sic) for > transcendentals? I did not use the word transcendental I think? And I am so sorry that English is not my native language. > Are you suggesting the sequence itself is finite? Any decimal representation of a real number terminates if ond only if it is rational and has (in its simplest expression) a denominator that contains only the primes 2 and 5. With sequence of finite terminations I understand the sequence where the first element is the decimal expansion terminated after the first digit, etc. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion >>Please quote what I said about 'cardinality' in general terms that is >>not handled by my definition. >> Cardinality(x)=least ordinal(y) with equinumerosity(z). I didn't post that or anything else nonsensical like that. Please don't >do that. You mean that mathematical definitions can't have different domains of discourse and mathematical definitions in different domains of discourse can't borrow from one another? Uncommonly curious I must say. Then of course one must further focus on definition of the domain itself. I mean how can a mathematical definition be complete without a further definition of x such that we need a definition for cardinality that ran Card(xd(y) . . . it would appear more or less indefinitely. Why just today in reply to a collateral post tells some one that Stephen has defined sets IN and OUT when we all realize that actual mathematical definitions would require IN(x) and OUT(x) to be defined instead. Now as to whether I should post such constructions in reply to you, I think you'll find ironic hyperbolic rhetoric is a perfectly acceptable forensic modality. If you can seriously suggest I should quote you on subjects that you can't even be bothered to research and justify for yourself I'll reply in any way I ing well please which extends a legitimate avenue of discussion whether you like it or not, Moe. You asked for a quote and I posted a possible reinterpretation in reply which may not have been responsive to your demand but is certainly a legitimate implication of what you've had to say on the subject. ~v~~ === Subject: Re: Cantor Confusion <15nvj2lva8o0e01qnlsqcjvm5veepl7itq@4ax> <7p22k21t56iik4bjuptedo9cii5jpnkiqp@4ax> <17e2k2hbv56egdkh4gnnlbh9uusij787qh@4ax> >Please quote what I said about 'cardinality' in general terms that is >>not handled by my definition. >> Cardinality(x)=least ordinal(y) with equinumerosity(z). >I didn't post that or anything else nonsensical like that. Please don't >do that. You mean that mathematical definitions can't have different domains > of discourse and mathematical definitions in different domains of > discourse can't borrow from one another? No, that's not what I said. > Now as to whether I should post such constructions in reply to you, I > think you'll find ironic hyperbolic rhetoric is a perfectly acceptable > forensic modality. If you can seriously suggest I should quote you on > subjects that you can't even be bothered to research and justify for > yourself I'll reply in any way I ing well please which extends a > legitimate avenue of discussion whether you like it or not, Moe. You > asked for a quote and I posted a possible reinterpretation in reply > which may not have been responsive to your demand but is certainly a > legitimate implication of what you've had to say on the subject. No, you posted utter nonsense (Cardinality(x)=least ordinal(y) with equinumerosity(z)) as if it is something that I had said. That is not ironic hyperbolic rhetoric nor a legitimate implication of what I said. MoeBlee === Subject: Re: Cantor Confusion >Please quote what I said about 'cardinality' in general terms that is >not handled by my definition. >> Cardinality(x)=least ordinal(y) with equinumerosity(z). >>I didn't post that or anything else nonsensical like that. Please don't >>do that. >> You mean that mathematical definitions can't have different domains >> of discourse and mathematical definitions in different domains of >> discourse can't borrow from one another? No, that's not what I said. Then why exactly are you complaining about what I said? Frankly, Moe, you don't seem to have said much of anything that I can make out. If mathematical definitions can have different domains of discourse then definition of mathematical definitions and domains of discourse.. >> Now as to whether I should post such constructions in reply to you, I >> think you'll find ironic hyperbolic rhetoric is a perfectly acceptable >> forensic modality. If you can seriously suggest I should quote you on >> subjects that you can't even be bothered to research and justify for >> yourself I'll reply in any way I ing well please which extends a >> legitimate avenue of discussion whether you like it or not, Moe. You >> asked for a quote and I posted a possible reinterpretation in reply >> which may not have been responsive to your demand but is certainly a >> legitimate implication of what you've had to say on the subject. No, you posted utter nonsense (Cardinality(x)=least ordinal(y) with >equinumerosity(z)) as if it is something that I had said. I never said you had said that. I see nothing utter or nonsensical about what I said. It seems pretty consistent with what you have said especially considering your disclaimer above regarding what you didn't say, I mean if you can follow all the logic involved. > That is not >ironic hyperbolic rhetoric nor a legitimate implication of what I >said. Says who exactly, Moe? Who died and made you arbiter of the universe? ~v~~ === Subject: Re: Cantor Confusion <15nvj2lva8o0e01qnlsqcjvm5veepl7itq@4ax> <7p22k21t56iik4bjuptedo9cii5jpnkiqp@4ax> <17e2k2hbv56egdkh4gnnlbh9uusij787qh@4ax> <4p35k2hco3ufon8m0tngbskrdf3ni7f0f6@4ax >> You mean that mathematical definitions can't have different domains >> of discourse and mathematical definitions in different domains of >> discourse can't borrow from one another? >No, that's not what I said. Then why exactly are you complaining about what I said? Frankly, Moe, > you don't seem to have said much of anything that I can make out. If > mathematical definitions can have different domains of discourse then > definition of mathematical definitions and domains of discourse.. Since I never said anything that can be paraphrased as the jumble of nonsense you just mentioned, nothing I did write entails that the >No, you posted utter nonsense (Cardinality(x)=least ordinal(y) with >equinumerosity(z)) as if it is something that I had said. I never said you had said that. You're absurd. You quoted me asking you what I said that justified a certain statement you made. You directly replied to that quote with Cardinality(x)=least ordinal(y) with >equinumerosity(z). > Says who exactly, Moe? Who died and made you arbiter of the universe? Apparently a rival of the authority that died and made you the arbiter as to what is perfectly acceptable forensic modality. MoeBlee P.S. My doubleposts and duplicate passages (from posts I thought were not previously accepted by the interface) are unintentional. === Subject: Re: Cantor Confusion > You mean I should disregard silence on your part as a preference for >> my not justifying your own opinions on the subject of your beliefs? Yes. Exactly. You hold the key to all that is true. Of course. I'd have mentioned it myself but I didn't want to put too fine a point on the obvious. ~v~~ === Subject: Re: Cantor Confusion What you regard as foolish is the explanation of the axioms which seem > to be your gospel. These axioms and their meaning have not yet changed > (as far as I know from modern text books and from the internet page of > T. Jech (a leading set theorist of our days)). Which modern text book have you read? I cannot find any non-biographical texts on Jech's internet page. Please do elaborate (or rather: please don't...) === Subject: Re: Cantor Confusion >> What you regard as foolish is the explanation of the axioms which seem >> to be your gospel. These axioms and their meaning have not yet changed >> (as far as I know from modern text books and from the internet page of >> T. Jech (a leading set theorist of our days)). Which modern text book have you read? I cannot find any > non-biographical texts on Jech's internet page. Please do elaborate (or > rather: please don't...) I have to correct myself here: I have found an archive of published text. Perhaps I'll comment on them later. === Subject: Re: Cantor Confusion >> What you propose, namely the infinity of ZF without the axiom INF would >> not be an advance. But meanwhile you may have recognized that your >> assertion (ZF even without INF is not finite) is false. > > It is, however, quite true that ZF without INF need not be finite. It is, more than that, quite true that ZF without INF _is_ infinite (the axiom schema of separation alone provides infinitely many axioms). The point is: ZF without INF does not prohibit the existence of infinite sets, nor does it force them to exist. === Subject: Re: Cantor Confusion > >>All the balls have been removed before noon. >>OK. >But more balls are in the vase. >>Reason? Proof? Example? Anything? >>Consider a strictly increasing sequence with non-negative >>terms.--------- If you can. >Consider that the number of balls as a function of time has infinitely > many integer jump discontinuities which cluster around noon, so that > there is no way that the function can be continuous at noon. Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a vase by the water molecules in an ocean - what hell is the difference!? Then use a continuous model, as is _routinely done_ with Fluid Dynamics. And there IS a way that the function can be continuous at noon. But the problem is that you mathematicians do not understand what continuity IS. You cannot comprehend that there can be a discrete as well as continuous description for one and the same (physical) phenomenon. See for example the Fluid Tube Continuum: http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft Han de Bruijn === Subject: Re: Cantor Confusion >All the balls have been removed before noon. >>OK. >But more balls are in the vase. >>Reason? Proof? Example? Anything? >>Consider a strictly increasing sequence with non-negative >>terms.--------- If you can. >Consider that the number of balls as a function of time has infinitely > many integer jump discontinuities which cluster around noon, so that > there is no way that the function can be continuous at noon. >Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a > vase by the water molecules in an ocean - what hell is the difference!? > Then use a continuous model, as is _routinely done_ with Fluid Dynamics. > And there IS a way that the function can be continuous at noon. But the > problem is that you mathematicians do not understand what continuity IS. > You cannot comprehend that there can be a discrete as well as continuous > description for one and the same (physical) phenomenon. See for example > the Fluid Tube Continuum: >http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft >Han de Bruijn Does HdB claim that the vase problem can be run in the physical world? What the Hell is the difference is that in mathematical worlds things do not always conform to the demands of physicists. Does HdB dispute that, according to the Gedankenexperiment as originally stated that, every ball inserted before noon is also removed before noon? === Subject: Re: Cantor Confusion >> >>All the balls have been removed before noon. >>OK. >But more balls are in the vase. >>Reason? Proof? Example? Anything? >>Consider a strictly increasing sequence with non-negative >terms.--------- If you can. >> >> Consider that the number of balls as a function of time has infinitely >> many integer jump discontinuities which cluster around noon, so that >> there is no way that the function can be continuous at noon. > Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a > vase by the water molecules in an ocean - what hell is the difference!? > Then use a continuous model, as is _routinely done_ with Fluid Dynamics. > And there IS a way that the function can be continuous at noon. But the > problem is that you mathematicians do not understand what continuity IS. Irrelevant insult noted. > You cannot comprehend that there can be a discrete as well as continuous > description for one and the same (physical) phenomenon. See for example > the Fluid Tube Continuum: Can you describe a continuous version of the problem where each unit of water has a well defined exit time? A key part of the original problem is that the time at which each ball is removed is defined and reached. This is crucial to the problem. It is not just a matter of rates. If you added balls 1-10, then 2-20, 3-30, ... but you removed balls 2,4,6,8, ... then the vase is not empty at noon, even though the rates of insertions and removals are the same as in the original problem. So you cannot just say the rate is 10 in and 1 out and base an answer on that. Another key feature of the problem is that there are truly an infinite number of balls. In the infinite case it is true that 1) you add 10 balls and remove one ball at each step 2) each ball has a specified time of removal, and is removed at that time. In a finite version, you can only satisfy one of those conditions. Which one you pick determines the answer. Stephen === Subject: Re: Cantor Confusion > Consider that the number of balls as a function of time has infinitely > many integer jump discontinuities which cluster around noon, so that > there is no way that the function can be continuous at noon. >Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a > vase by the water molecules in an ocean - what hell is the difference!? > Then use a continuous model, as is _routinely done_ with Fluid Dynamics. > And there IS a way that the function can be continuous at noon. So, you are saying that if we change the problem, we can get your answer. I don't think anyone doubts this. Not sure why you think anyone would doubt it. > But the > problem is that you mathematicians do not understand what continuity IS. > You cannot comprehend that there can be a discrete as well as continuous > description for one and the same (physical) phenomenon. I guess you missed the point that the ball and vase problem is not a physical problem. The infinite number of balls should have been a hint. -- David Marcus === Subject: Re: Cantor Confusion > Please name the mathematicians that agree that your argument is correct. > (Han doesn't count, since he says he is a physicist.) A Theoretical Physicist. Mathematical Physics and Physical Mathematics. Han de Bruijn === Subject: Re: Cantor Confusion > [ ... snip ... ] By the way, the first draft of my website > on MatheRealism is ready including links to your site. >http://www.fh-augsburg.de/~mueckenh/MR.mht I've seen it. Very good! Han de Bruijn === Subject: Uniform integrability Let (X,M,mu) be a positive measure space and S a subset of L^1(mu). I saw two definitions of uniform integrability. Definition 1: S is uniformly integrable if to each epsilon>0 corresponds a delta>0 such that | int_E f dmu | < epsilon whenever f in S and mu(E) Let (X,M,mu) be a positive measure space and S a > subset of L^1(mu). I saw > two definitions of uniform integrability. >Definition 1: S is uniformly integrable if to each > epsilon>0 corresponds a > delta>0 such that | int_E f dmu | < epsilon >whenever f in S and mu(E)Definition 2 replaces | int_E f dmu | above by int_E > |f| dmu. >Can anyone give an example to show that these two > definitions are not > equivalent? (I think they are not equivalent.) > I think they are equivalent. Any S satisfying definition 2 obviously satisfies 1. Take S to satisfy 1. Let epsilon be given, and delta be as in definition 1 except where we replace epsilon by epsilon/2 (so |int_E f dmu| Let (X,M,mu) be a positive measure space and S a >subset of L^1(mu). I assume that each f in S is complex-valued. >I saw two definitions of uniform integrability. Definition 1: S is uniformly integrable if to each epsilon>0 corresponds a > delta>0 such that | int_E f dmu | < epsilon whenever f in S and mu(E) equivalent? (I think they are not equivalent.) === Subject: Re: Uniform integrability >Let (X,M,mu) be a positive measure space and S a >subset of L^1(mu). >I assume that each f in S is complex-valued. > >I saw two definitions of uniform integrability. > Definition 1: S is uniformly integrable if to each > epsilon>0 corresponds a > delta>0 such that | int_E f dmu | < epsilon > whenever f in S and mu(E) Definition 2 replaces | int_E f dmu | above by > int_E |f| dmu. > Can anyone give an example to show that these two > definitions are not > equivalent? (I think they are not equivalent.) Didn't see the complex valued thing before, although I think what I said in the last post could be extended to the complex case. In particular note: |f(x)| < c(|Re(f(x)|+|Im(f(x))|) for some constant c by equivalency of norms. === Subject: JSH: Why factoring idea changed I had a simple idea for factoring which I felt really should work, and I'd done quite a bit of analysis with it. But I had this variable S, which I finally noticed took away the likelihood that the idea DID actually work. So I thought a bit about how to get rid of it and came up with a slightly different idea: x^2 - y^2 = 0 mod T and k^2 = 2xk mod T so now it's (x+k)^2 = y^2 + 2k^2 + nT where I introduce n, which has to be nonzero, and you pick n and k, where the simplest thing may be to let n=-1, and then find the smallest k such that 2k^2 - T is positive to give a small number to factor as you factor 2k^2 + nT to find y and then x. By tossing S, I bring the variable count down and make it where the math, hopefully, will realize what your target T is, while before, with S, it could be just about anything. === Subject: Re: JSH: Why factoring idea changed > I had a simple idea for factoring which I felt really should work, In other words, you never _proved_ it would work. > and > I'd done quite a bit of analysis with it. But I had this variable S, > which I finally noticed took away the likelihood that the idea DID > actually work. In other words, you _couldn't_ prove it would work. So I thought a bit about how to get rid of it and came up with a > slightly different idea: x^2 - y^2 = 0 mod T and k^2 = 2xk mod T so now it's (x+k)^2 = y^2 + 2k^2 + nT where I introduce n, which has to be nonzero, and you pick n and k, > where the simplest thing may be to let n=-1, and then find the smallest > k such that 2k^2 - T is positive to give a small number to factor as you factor 2k^2 + nT to > find y and then x. By tossing S, I bring the variable count down and make it where the > math, hopefully, will realize what your target T is, Hope isn't proof. > while before, with > S, it could be just about anything. === Subject: Re: JSH: Why factoring idea changed > I had a simple idea for factoring which I felt really should work, How about proving something of consequence instead of polluting the place with this sillyness ? http://sciphysicsopenmanuscript.blogspot/ Prove that the existence of a trivial is arbitrary, or indeterminate. That there is no way to determine if an object is really itself, or if it is an exact clone of itself, that such a thing is indeterminate. Should be pretty easy. I found your yearbook photo >http://sciphysicsopenmanuscript.blogspot/ === Subject: Re: JSH: Why factoring idea changed >I had a simple idea for factoring which I felt really should work, How about proving something of consequence instead of polluting the place > with this sillyness ? Why are you responding to mensanator with this? === Subject: Re: JSH: Why factoring idea changed I had a simple idea for factoring which I felt really should work, > > How about proving something of consequence instead of polluting the place > with this sillyness ? Why are you responding to mensanator with this? The sci.math readership is not very bright. Besides the idea IS a solution which astute readers will realize is equivalent to using S = k^2 with the earlier equation. So what I had before was a general factoring solution--but too general. With S disconnected from k, there was nothing locking the equations to your target T, so the math could range all over the place at will. Now by using S = k^2 you in a sense tell the math that your target is T, when you use nonzero n. The mathematician con game is almost over. And it has been a con game where the cons were too stupid to know when to give up, or they wouldn't have been fighting my research, forcing me to find something they couldn't just lie about, so here we are. The future of humanity WAS in the balance. === Subject: Re: JSH: Why factoring idea changed I had a simple idea for factoring which I felt really should work, > > How about proving something of consequence instead of polluting the place >with this sillyness ? > Why are you responding to mensanator with this? The sci.math readership is not very bright. Besides the idea IS a solution which astute readers will realize is > equivalent to using S = k^2 with the earlier equation. So what I had before was a general factoring solution--but too general. With S disconnected from k, there was nothing locking the equations to > your target T, so the math could range all over the place at will. Now by using S = k^2 you in a sense tell the math that your target is > T, when you use nonzero n. The mathematician con game is almost over. And it has been a con game > where the cons were too stupid to know when to give up, or they > wouldn't have been fighting my research, forcing me to find something > they couldn't just lie about, so here we are. The future of humanity WAS in the balance. > Deja vu? I could have sworn you had already stated that you had won the math wars less than a week ago. Yet you were wrong. Just like the couple dozen times before that. If I was a betting man, I would say you were wrong again. And looking at your math, I would definitely take that bet. === Subject: Re: JSH: Why factoring idea changed I had a simple idea for factoring which I felt really should work, > >How about proving something of consequence instead of polluting the place >with this sillyness ? > Why are you responding to mensanator with this? > The sci.math readership is not very bright. > Besides the idea IS a solution which astute readers will realize is > equivalent to using > S = k^2 > with the earlier equation. > So what I had before was a general factoring solution--but too general. > With S disconnected from k, there was nothing locking the equations to > your target T, so the math could range all over the place at will. > Now by using S = k^2 you in a sense tell the math that your target is > T, when you use nonzero n. > The mathematician con game is almost over. And it has been a con game > where the cons were too stupid to know when to give up, or they > wouldn't have been fighting my research, forcing me to find something > they couldn't just lie about, so here we are. > The future of humanity WAS in the balance. > > Deja vu? I could have sworn you had already stated that you had won the math > wars less than a week ago. Yet you were wrong. Just like the couple > dozen times before that. If I was a betting man, I would say you were wrong again. And looking > at your math, I would definitely take that bet. Your identity will soon be known. === Subject: Re: JSH: Why factoring idea changed > sounds like a fortune cookie. Are you using your new modified corrected, recorrected non-S factoring idea to find out this information ? Just factor the front part of this his message tag, and you got him! you can drive over to his house and show, explain how your factoring figures all of this out, and how you saved the world. Then you can try to find out who I am, a true Pure Mathematician and stop by my house, and I laugh in your face, monkey boy! === Subject: Re: JSH: Why factoring idea changed I had a simple idea for factoring which I felt really should work, >How about proving something of consequence instead of polluting the place >> with this sillyness ? >Why are you responding to mensanator with this? > The sci.math readership is not very bright. > Besides the idea IS a solution which astute readers will realize is >equivalent to using > S = k^2 > with the earlier equation. > So what I had before was a general factoring solution--but too general. > With S disconnected from k, there was nothing locking the equations to >your target T, so the math could range all over the place at will. > Now by using S = k^2 you in a sense tell the math that your target is >T, when you use nonzero n. > The mathematician con game is almost over. And it has been a con game >where the cons were too stupid to know when to give up, or they >wouldn't have been fighting my research, forcing me to find something >they couldn't just lie about, so here we are. > The future of humanity WAS in the balance. > >Deja vu? > I could have sworn you had already stated that you had won the math > wars less than a week ago. Yet you were wrong. Just like the couple > dozen times before that. > If I was a betting man, I would say you were wrong again. And looking > at your math, I would definitely take that bet. Your identity will soon be known. > Please, post it as soon as possible. I suspect that you won't, since everything you try, you fail at. As far as the police or other authorities, I live in the real world and have nothing to fear from them. === Subject: Re: JSH: How would you prove a corrupted math field? >jstevh@msn skrev: >> Here's a what if, as what if the mathematical community figured out it >> didn't have to have results that actually worked in pure math areas >> as all that is necessary is to have other mathematicians AGREE that a >> result worked. >Then mathematicians could just keep each other working in those areas >> without ever bothering with having real results, and who could stop >> them? >So all grad students obey like zombies, and you are the first person in >history to speak out? >--- >J K Haugland >http://home.no.net/zamunda > Well, going from the hypothetical--remember the start of the post is to >imagine a what if...what if the math field were corrupted in pure >math areas, how would you prove it--to my experiences I had a math >grad student from Cornell contact me by email himself. > Seems he had noticed the arguing on Usenet and offered to help out, >asking me to explain my non-polynomial factorization argument to him, >so I did, and as I gave him pieces, he re-worked them in his own words. > For some reason (hmmm) it took him MONTHS to go through an entire >argument, until he is at the end and I am thinking, hey, maybe a math >grad will do the right thing, and he claims that he can see how it is >true with INTEGERS but is not sure about algebraic integers and he >needs to go study up on them. > Needless to say, nothing else substantive from that guy, > Could it be that this guy lost a bet? As a result he had to try and > befriend you? > > and really why I don't know why you stopped there, for readers who wonder if I cut him > off as I didn't. Who knows exactly why the Cornell grad student contacted me, though I > can tell you what he said which wasn't what he did so it hardly > matters, but it goes to the back work that I did BEFORE my paper was So I had this student working through all the major areas in his own > words but begging off the conclusion. I had the arguments before on > Usenet which I DID consider carefully. And I had an in-person > conference with a math professor at my alma mater Vanderbilt > University. So I did a tremendous amount of due diligence and had quite a lot of > limited feedback from Barry Mazur, and even less but still some in a > way from Andrew Granville. Yes the paper did have some minor mistakes on my part like grammar, but > nothing that changed the conclusion, which is that there is this > problem with algebraic integers. So the math grad student from Cornell is real world proof that the idea > that grad students would come out against--their bosses--full > professors is just stupid. They follow along just as closely or more > closely as their CAREER is on the line. A grad student coming out against professors can expect to lose and > lose a lot, so they are the worst people to go to with some new idea > that professors are dodging as they have even less power. Your one hope is to get backup from a full professor, who has > department backup or you're screwed. Even an individual mathematician > is worthless against the group. You need a professor, a full professor, and you need him to get backup > from his department first, if there is any chance at all. What you _really_ need is to be correct. === Subject: Re: JSH: How would you prove a corrupted math field? >jstevh@msn skrev: >> Here's a what if, as what if the mathematical community figured out it >> didn't have to have results that actually worked in pure math areas >> as all that is necessary is to have other mathematicians AGREE that a >> result worked. >Then mathematicians could just keep each other working in those areas >> without ever bothering with having real results, and who could stop >> them? >So all grad students obey like zombies, and you are the first person in >history to speak out? >--- >J K Haugland >http://home.no.net/zamunda > Well, going from the hypothetical--remember the start of the post is to >imagine a what if...what if the math field were corrupted in pure >math areas, how would you prove it--to my experiences I had a math >grad student from Cornell contact me by email himself. > Seems he had noticed the arguing on Usenet and offered to help out, >asking me to explain my non-polynomial factorization argument to him, >so I did, and as I gave him pieces, he re-worked them in his own words. > For some reason (hmmm) it took him MONTHS to go through an entire >argument, until he is at the end and I am thinking, hey, maybe a math >grad will do the right thing, and he claims that he can see how it is >true with INTEGERS but is not sure about algebraic integers and he >needs to go study up on them. > Needless to say, nothing else substantive from that guy, > Could it be that this guy lost a bet? As a result he had to try and > befriend you? > > and really why I don't know why you stopped there, for readers who wonder if I cut him > off as I didn't. Who knows exactly why the Cornell grad student contacted me, though I > can tell you what he said which wasn't what he did so it hardly > matters, but it goes to the back work that I did BEFORE my paper was So I had this student working through all the major areas in his own > words but begging off the conclusion. I had the arguments before on > Usenet which I DID consider carefully. And I had an in-person > conference with a math professor at my alma mater Vanderbilt > University. So I did a tremendous amount of due diligence and had quite a lot of > limited feedback from Barry Mazur, and even less but still some in a > way from Andrew Granville. Yes the paper did have some minor mistakes on my part like grammar, but > nothing that changed the conclusion, which is that there is this > problem with algebraic integers. So the math grad student from Cornell is real world proof that the idea > that grad students would come out against--their bosses--full > professors is just stupid. They follow along just as closely or more > closely as their CAREER is on the line. A grad student coming out against professors can expect to lose and > lose a lot, so they are the worst people to go to with some new idea > that professors are dodging as they have even less power. Your one hope is to get backup from a full professor, who has > department backup or you're screwed. Even an individual mathematician > is worthless against the group. You need a professor, a full professor, and you need him to get backup > from his department first, if there is any chance at all. > Though a failure at mathematics you still excell at being a ranting tit I see. === Subject: Re: JSH: How would you prove a corrupted math field? courses, into Analysis, Abstract Algebra, and the like, the class > stops being The professors states a theorem and tells you to believe > it. Theorems are ->proven<-, in detail. Students are required to > check and verify those proofs; in fact, they are required to attempt > to challenge them in any and all ways possible. Students don't just > get told what Galois Theory says (or how it is interpreted); they > aren't just told what Dedekind did with ideals. The results are > ->established<- and ->proven<-. The verification of those results has > never been an appeal to authority, but has always been done by > generation upon generation of students, upper division and graduate, > who are required to verify those results step by step and link by > link. I used to write some fiction, and the initial post read to me like a fishing expedition for a story idea. The backdrop quoted above can be used for the kind of story that scenarioized about by a fiction writer with some inventiveness: all it would take for such a conspiracy to get ahold of American math would be a kind of distractive-monitor committee to veer the students away from the kind of math book(s) that would expose the fallacious groundwork behind conventionally-accepted math, and to quietly get rid of a student who was inquisitive in the wrong way. The trouble with such a story, though, would be that the only believable hero in it - the savior of math - would have to be someone who came from that world, and thrived in it at the grad-school level, but who got eased out because he, or she, was veering in on that forbidden truth. It would have to be a protagonist like Helmholtz Watson of BRAVE NEW WORLD, or the protagonist of John Brunner's THE SHOCKWAVE RIDER. In addition, such a person would have to have been a character development would have to be a psychological struggled tied in with the difficulty of seeing through the collective blind spot. [I can say with some confidence that it would take years, even for a minor flaw, because the flaws that haven't been found out already would have to be subtle ones.] Making it an emperor-has-no-clothes story would render it unbelievable, except as a kind of fictionalized tract. === Subject: Re: JSH: How would you prove a corrupted math field? >> How would a person go about proving that the field was corrupted ...? > First, one must prove that the existence of a trivial is arbitrary, or indeterminate. http://sciphysicsopenmanuscript.blogspot/ James - I found your yearbook picture from highschool. It's on my log --->> http://sciphysicsopenmanuscript.blogspot/ === Subject: Re: JSH: How would you prove a corrupted math field? > jstevh@msn skrev: >Here's a what if, as what if the mathematical community figured out it >didn't have to have results that actually worked in pure math areas >as all that is necessary is to have other mathematicians AGREE that a >result worked. > Then mathematicians could just keep each other working in those areas >without ever bothering with having real results, and who could stop >them? > So all grad students obey like zombies, and you are the first person in > history to speak out? > --- > J K Haugland > http://home.no.net/zamunda Well, going from the hypothetical--remember the start of the post is to > imagine a what if...what if the math field were corrupted in pure > math areas, how would you prove it--to my experiences I had a math > grad student from Cornell contact me by email himself. Seems he had noticed the arguing on Usenet and offered to help out, > asking me to explain my non-polynomial factorization argument to him, > so I did, and as I gave him pieces, he re-worked them in his own words. For some reason (hmmm) it took him MONTHS to go through an entire > argument, Probably because he was unsure what JSH meant by some terms. I've run across this before; earlier this year, I had to compile an ArchimedesPlutonium-English dictionary to properly understand what he was talking about. It involves some guesswork as well. Just imagine someone painting hieroglyphics in a room, and then being told that it will take thousands of years before anyone who speaks English will know what it took him only five minutes to write. Put yourself in other people's shoes for once, JSH! > until he is at the end and I am thinking, hey, maybe a math > grad will do the right thing, and he claims that he can see how it is > true with INTEGERS but is not sure about algebraic integers and he > needs to go study up on them. Needless to say, nothing else substantive from that guy, He was reporting JSH to the NSA. > and really why > would any rational adult who understands the real world think that grad > students would speak out, potentially trash their careers, What career? Next thing you know, JSH will be claiming that _high school students_ have careers ... --- Christopher Heckman > when being quiet they can just go with the status quo, and get their degree? As an analogy, consider again what happened in America with the run-up > to war with Iraq and there LIVES were on the line. Americans were > either totally swalloing the party line of George Bush and company or > were TERRIFIED to speak out, and they had examples of people who faced > retribution when they tried to exercise their freedom of speech. Where > the Dixie Chicks were just one of the more famous examples. Here in the land of the free and the home of the brave the brave got > the crap kicked out of them when they spoke out against what most > American wanted to believe, and people said that was just about > patriotism. Gee, who would a thunk that trashing American values, making us look > sick and stupid around the world, so we could go kill some people in > some weaker country was really patriotic? Groups are notorious for making the wrong decisions and people part of > groups are typical in NOT going against the group opinion. So no, math > grads are not a defense against the scenario I painted. === Subject: Re: How would you prove a corrupted math field? > Here's a what if... > How would a person go about proving that the field was corrupted ...? > You need to use corrupt field theory. This is similar to regular field > theory, except corrupted rules are used. For example: Instead of associativity, use asocial-activity. > Instead of commutativity, use comic-activity. > Instead of distributivity, use disturb-activity. > Instead of division rings, use derision rings. > and so on... It's likely that some newsgroup posters are experts in corrupt field > theory. Carl G. > JSH has his own theory about the distributive theory and also the multiplies through theory I am sure he has spent many many hours on these new breakthroughs. === Subject: Re: JSH: How would you prove a corrupted math field? mathematics in the classroom never rose above a standard calculus > course. As such, his experience would be limited to a professor ->stating<- a > result, giving plausibility arguments, and never actually ->proving<- > the result. By way of example, very few, if any, calculus textbooks in the > U.S. contain a proof of the Intermediate Value Theorem, though all of > them include the theorem. They just say it is clear. (Giving a truly > rigorous proof is easy if you accept the Supremum Principle, though of > course it is equivalent to that; the Sup. Principle can be rigorously > proved using Dedekind cuts, for example). Students are given the > Theorem, and told to use it. The same is true for any number of > results covered in Calculus. Now, of course, Calculus and lower-division linear algebra are service > courses. In particular, Calculus is not and should not be an analysis > class. It is about, as the name implies, calculations; using > calculus. Lower division linear algebra is not about the beauty of > abstraction and abstract algebra, proceding from axioms to great > theorems, but about solving systems of linear equations and > differential equations, about R^n, about least squares, etc. So there > is good reason for that. This was very much my experience with my undergraduate physics degree here in the UK, and I guess it's the same in the States. By the time I finished my MPhys I really had no idea what a rigorous proof looked like, so I imagine your speculation is probably spot-on. -Rotwang === Subject: Re: JSH: How would you prove a corrupted math field? gjedwards@gmail suggested: > Factor a big number. Glad to be of help. Hmm. Long ago I was given the task of reviewing a math dept. submission from a couple of folks at a community college who had figured out how to factor arbitrarily large numbers... of the form 10^n. They wanted to know if we were interested in a license agreement. No, I'd rather see someone of JSH's interests tell us precisely how many primes there are between 1 and 10^23. === Subject: Re: JSH: How would you prove a corrupted math field? > [jstevh@msn] >> Here's a what if, as what if the mathematical community figured out it >> didn't have to have results that actually worked in pure math areas >> as all that is necessary is to have other mathematicians AGREE that a >> result worked. >> Then mathematicians could just keep each other working in those areas >> without ever bothering with having real results, and who could stop >> them? >> What if? Any idea how that scenario could be blocked? >> How would a person go about proving that the field was corrupted if >> every top mathematician was just going to say no, and only >> mathematicians would chime in to protect, where the money aspect is >> HUGE with grants and prizes considered, and human nature being what it >> is, why couldn't unsavory individual infiltrate the discipline, get a >> critical mass together, and then just take over? >> It'd be a con's dream--a perfect situation with hard to understand >> information, a cowed world in awe of the field, and no need to ever >> produce results that actually worked as hey, they could just call them >> pure!!! >> How could anyone break through if such a scenario were true? First thing I'd do is pay attention to my nose, very carefully. After > noticing the persistent odor of popcorn, it just may occur to me that I > must have fallen asleep during the movie, and it's all just a dream. > Sometimes that's enough of a shock to wake up. But sometimes not -- the > popcorn can be worked into the dream too. Then the only hope is to notice that the stories I'm telling myself get > increasingly silly as time goes on. For example, I might consider that > anyone determined to get rich has a much better chance studying to be a > rodeo clown than going into pure math. Terence Tao is getting the > Ramanujan Prize this year! One thing onlookers aren't speculating about > is whether he'll retire on the US$10,000 prize money. Maybe that's HUGE in your dream, though. If so, that's also one of the > signs of a dream: /many/ things in a dream are bizarre. > JSH is self-blocking, he can not seem to realize it, and remains success challenged. This reminds me of two people that overcame this and became famous. Before I was shot, I always thought that I was more half-there than all-there - I always suspected that I was watching TV instead of living life.Right when I was being shot and ever since, I knew that I was watching television. The channels switch, but it's all television. -- Andy Warhol Revolutionaries must firmly establish a revolutionary outlook on the leader. To acquire a revolutionary outlook on the leader means to devote oneself to the leader, with a correct understanding that the leader is the centre of the socio-political organism, and hold him in high esteem with a === Subject: Re: JSH: Go with prime counting > When it comes to the question of whether or not the math field IS > corrupted in pure math you can answer that question easily by going > to prime counting. Primes are just such a popular area and they are such a big deal in > mathematics that it's not hard if you want to get to the bottom of > things to evaluate claims in that area, and denials. The math world I thought existed before I had my own math discoveries > would not have ignored, dismissed or downplayed my prime counting > research. But cons pretending to be real mathematicians who care about > mathematics would, if it suited them. That's what they do. Cons do what suits them, not what's important to > the world. Here they have turned the system upside down, where legitimate research > is ignored, while their made up research gets their people prizes, > grants, and accolades. That's how they can ignore research on primes, and how they can ignore > simple factoring ideas like what I've talked about recently. Go with prime counting. Just do a web search on the subject, look at > previous research and even notice how similar they are to what I > have--and then note the differences. Used to be you might have believed mathematicians cared so much about > mathematics they'd chase down every detail, get excited about every > little thing that might be new in an important area--but notice how > that goes away with someone like me--a harsh critic willing to tell you > these people lie. Do the web search. Easy: prime counting. Go do it. to leave on Google Groups. === Subject: The Fundamentals of Trinary Group Theory (was: trinary group theory?) trinary group theory? sci.math Is there such a thing as trinary group theory? A group can more or less be described as a two- > dimensional multiplication table (finite groups at least). > My question is, is there a type of group that has a > three-dimensional multiplication table? In other words, is there a group theory that describes > trinary operations, instead of binary operations? Yes, a group without the identity. In place of the operation ab is the translation covariant operation a/b.c = a b^{-1} c. As you may verify, this operation is covariant with respect go the map x |-> gxh. That is, (gah) (gbh)^{-1} (gch) = g (a b^{-1} c) h. So, it qualifies as a bona fide forgot the location of the identity operation. Suitable axioms are almost trivial to write down: a/a.b = b; a/b.b = a; a/b.(c/d.e) = (a/b.c)/d.e. For an abelian member of this family, one also have the axiom a/b.c = c/b.a. Picking an element e of the set, one can then proceed to define relative group operations by ab = a/e.b; a^{-1} = e/a.e, and easily prove this is a group. In fact, the group for e is isomorphic to the group for e' with the mapping x |-> x/e.f. Calling the system T, the group dT associated with T can be defined intrinsically. Take ordered pairs out of T x T and pose the equivalence relation generated by the identity (a, b/c.d) = (c/b.a, d). Then define ab to be the corresponding equivalence class [(a,b)]. Automatically, this operation satisfies the property a(b/c.d) = (c/b.a)d -- which may be taken as the definition of the product (ab)(cd). One easily proves that aa = bb for all a, b, allowing the identity to be defined as aa; and that (ab)(bc) = ac, and (ab)^{-1} = ba, thereby showing that dT is a group. Calling the group located at e, T_e, one proves that T_e is isomorphic to dT by either of the maps f_e: T_e -> dT; f_e(x) = ex g_e: dT -> T_e; g_e(ab) = e/a.b. Further, the group dT acts on T to the right by a (bc) = a/b.c. (One can reverse the order of all the operations and slashes to get a corresponding mirror image set of axioms and a left-action of the group dT on T). The ternary groups are called torsors or general affine spaces. === Subject: Re: Proof for an inequality I have been trying to prove the following inequality in the last few >days to no avail. I know it is true. >If someone helps, I appreciate. >Prove for a,b,c >=0 >6*(a^3+b^3+c^3)+9abc>=(a+b+c)^3 All terms in the inequality are homogeneous of degree 3 > in a,b,c, so it suffices to prove it on, say, the > simplex S = {(a,b,c): a+b+c = 1, a >= 0, b >= 0, c >= 0}. > Substituting c = 1-a-b, we want to prove f(a,b) >= 1 subject to constraints a >= 0, b >= 0, a + b <= 1 where f(a,b) = 6 (a^3+b^3+(1-a-b)^3) + 9ab (1-a-b). The Karush-Kuhn-Tucker conditions for minimizing f subject to > the conditions a >= 0, b >= 0, -a -b >= -1 are -18 a^2 + 18 (1-a-b)^2 - 9 b (1 - a - b) + 9 a b + t1 - t3 = 0 > -18 b^2 + 18 (1-a-b)^2 - 9 a (1 - a - b) + 9 a b + t2 - t3 = 0 > t1 a = 0 > t2 b = 0 > t3 (1-a-b) = 0 > t1 >= 0, t2 >= 0, t3 >=0, a >= 0, b >= 0, a+b <= 1 Using Maple, I find the equations have 10 solutions (it could > be done by hand, rather tediously). Of those, > only one satisfies the inequalities: t1 = t2 = t3 = 0, > a = b = 1/3. Since f(1/3, 1/3) = 1, that proves the minimum > value is 1, so the inequality is true. Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada I should thank you all for your responses. To Dr. Israel: This is a rather interesting coincidence because a couple of years ago I actually took your Linear Programming course. The materials in that course helped me understand your solution :) However this is one part that I dont quite get. How can we exapnd the proof given from the simplex to all non-negative a,b,c's? This is just for my curiosity and if you refer me to a link or a book is good enough. === Subject: Re: Proof for an inequality I have been trying to prove the following inequality in the last few >days to no avail. I know it is true. >If someone helps, I appreciate. >Prove for a,b,c >=0 >6*(a^3+b^3+c^3)+9abc>=(a+b+c)^3 > All terms in the inequality are homogeneous of degree 3 > in a,b,c, so it suffices to prove it on, say, the > simplex S = {(a,b,c): a+b+c = 1, a >= 0, b >= 0, c >= 0}. > Substituting c = 1-a-b, we want to prove > f(a,b) >= 1 subject to constraints a >= 0, b >= 0, a + b <= 1 > where f(a,b) = 6 (a^3+b^3+(1-a-b)^3) + 9ab (1-a-b). > The Karush-Kuhn-Tucker conditions for minimizing f subject to > the conditions a >= 0, b >= 0, -a -b >= -1 are > -18 a^2 + 18 (1-a-b)^2 - 9 b (1 - a - b) + 9 a b + t1 - t3 = 0 > -18 b^2 + 18 (1-a-b)^2 - 9 a (1 - a - b) + 9 a b + t2 - t3 = 0 > t1 a = 0 > t2 b = 0 > t3 (1-a-b) = 0 > t1 >= 0, t2 >= 0, t3 >=0, a >= 0, b >= 0, a+b <= 1 > Using Maple, I find the equations have 10 solutions (it could > be done by hand, rather tediously). Of those, > only one satisfies the inequalities: t1 = t2 = t3 = 0, > a = b = 1/3. Since f(1/3, 1/3) = 1, that proves the minimum > value is 1, so the inequality is true. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada I should thank you all for your responses. > To Dr. Israel: > This is a rather interesting coincidence because a couple of years ago > I actually took your Linear Programming course. The materials in that > course helped me understand your solution :) However this is one part > that I dont quite get. How can we exapnd the proof given from the > simplex to all non-negative a,b,c's? This is just for my curiosity and > if you refer me to a link or a book is good enough. If A = ta, B = tb, C = tc, then 6 (A^3 + B^3 + C^3) + 9 ABC = t^3 ( 6 (a^3+b^3+c^3)+9abc) and (A+B+C)^3 = t^3 (a+b+c)^3. That's what I mean by homogeneous of degree 3. So if t > 0, the inequality is true for (A,B,C) if and only if it is true for (a,b,c). In particular, take t = 1/(a+b+c) (where a,b,c >= 0 and not all are 0) and you get (A,B,C) in the simplex. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Proof for an inequality > Hi: I have been trying to prove the following inequality in the last few > days to no avail. I know it is true. > If someone helps, I appreciate. Prove for a,b,c >=0 > 6*(a^3+b^3+c^3)+9abc>=(a+b+c)^3 I did not have the patience to go through all responses, so I may be repeating someone's reply. Here it is: Without loss of generality, assume that the numbers are sorted: a >= b >= c >= 0 and we can introduce the non-negative differences b - c = d a - b = e so a = c + d + e b = c + d The expression whose non-negativity we need to prove is V = 6 * (a^3 + b^3 + c^3) + 9 * a * b * c - (a + b + c)^3 After the tedious substitution and simplification, we get (I confess to using symbolic algebra software) V = 9*c*d*e+9*c*d^2+9*c*e^2+6*d^2*e+12*d*e^2+4*d^3+5*e^3 which is visibly non-negative. Conditions for equality are also visible: d=e=0, that is, a=b=c. === Subject: Re: Proof for an inequality schrieb david petry : > This leads to another question: >We know that 9(a^3 + b^3 + c^3) >= (a+b+c)^3 > and we know 27abc <= (a+b+c)^3 >so we can ask, for what values of 't' is the following always true? >9(1-t)(a^3+b^3+c^) + 27 t abc >= (a+b+c)^3 >So far, we know it's true for 0 <= t <= 1/3 ... for non-negative values of a,b,c. > Does t=1/3 give the sharpest inequality? No, the inequality holds for all t <= 5/9. Proof by my favourite principle: Assume w.l.o.g. 0 <= a <= b <= c, so that you can write b = a + x c = a + x + y with nonnegative x and y; then expand 9(1-t)(a^3+b^3+c^3) + 27 t abc - (a+b+c)^3 into monomials in the variables a, x and y. - All coefficients of these monomials will be non-negative. This proves the inequality for t <= 5/9. On the other hand, if we substitute a := 0, b := 1, c := 1 into 9(1-t)(a^3+b^3+c^3) + 27 t abc - (a+b+c)^3, we obtain 2 (5 - 9t); so the inequality is sharp. By the way: The case t = 5/9, 4 (a^3+b^3+c^3)/3 + 5 abc >= (a+b+c)^3/3, is equivalent to Schur's inequality a^3+b^3+c^3 + 6 abc >= (a+b+c) (ab+bc+ca). === Subject: Did I stomp Maple? Did I stomp Maple (actually, its MATLAB adaptation)? I gave it a differential equation (D2z is the second derivative of z) (E1) (1-t)*D2z - (2+t)*Dz - 2*z = 0 and it gave me a lengthy answer involving Kummer functions of complex arguments. In fact, the two independent solutions are 1/(1-t)^2 and t*exp(-t)/(1-t)^2 directly checked. What were the designers doing? --------------------------- An explanation how I came across equation (E1): for my class, I started with solutions t and exp(t), set up a linear ODE with the above as a fundamental system, which is (E2) (1-t)*D2y + t*Dy - y = 0, then tested MATLAB to get the solutions back. It did, without a hitch. Then I posed a question (this time already not for the class): can we find an integrating factor z which turns the left side of (E2) into exact derivative? A way to do it is to solve the homogeneous ODE with the formally adjoint differential operator, and that is (E1). I solved it by factoring the operator -- the details are messy. And indeed, if you multiply (E2) by 1/(1-t)^2, you obtain D((Dy - y) / (1 - t)) = 0 as expected. Similarly with t*exp(t)/(1-t)^2. === Subject: Re: Did I stomp Maple? > Did I stomp Maple (actually, its MATLAB adaptation)? >I gave it a differential equation > (D2z is the second derivative of z) >(E1) (1-t)*D2z - (2+t)*Dz - 2*z = 0 >and it gave me a lengthy answer involving Kummer functions > of complex arguments. In fact, the two independent solutions are >1/(1-t)^2 and t*exp(-t)/(1-t)^2 >directly checked. >What were the designers doing? > Maple 10: E1 := (1-t)*diff(z(t),t,t) - (2+t)*diff(z(t),t) - 2*z(t) = 0; / d / d / d (1 - t) |--- |--- z(t)|| - (2 + t) |--- z(t)| - 2 z(t) = 0 dt dt // dt / dsolve(E1,z(t)); t _C1 _C2 z(t) = ---------------- + --------- 2 2 exp(t) (-1 + t) (-1 + t) But I did find this, with an extra t: E2 := (1-t)*diff(z(t),t,t) - (2+t)*diff(z(t),t) - 2*t*z(t) = 0; / d / d / d (1 - t) |--- |--- z(t)|| - (2 + t) |--- z(t)| - 2 t z(t) = 0 dt dt // dt / dsolve(E2,z(t)); / 1 / (1/2) /3 1 (1/2) z(t) = _C1 exp|- - t 1 + I 7 /| KummerM|- + -- I 7 , 3, 2 / 2 14 (1/2) I 7 (-1 + t)| / / 1 / (1/2) /3 1 (1/2) + _C2 exp|- - t 1 + I 7 /| KummerU|- + -- I 7 , 3, 2 / 2 14 (1/2) I 7 (-1 + t)| / -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: 10c - c = 9. This is the same as 9c = 9. Is that correct? >Do you not accept that there might be a commonly accepted default >meaning to 0.999...? >No, as evidenced by the number of threads on that > symbol. Without careful definition, there is no end > of confusion on some people's part as to whether it > stands for the sequence {0.9, 0.99, 0.999, ...} or > the limit of that sequence. >There must be some complete idiots who partake in such threads. === > I wouldn't know, I have /Subject:.*999.../ in my killfile. The matter can be handled better. Do not tell the original poster (troll) the mathematics. Demand that the op tell you exactly what he means by ASCII {0x2e, 0x39, 0x39, 0x39, 0x39, 0x39, 0x2e, 0x2e, 0x2e, 0x0a}. -- Michael Press === Subject: Re: 10c - c = 9. This is the same as 9c = 9. Is that correct? > Can Statement Number One {10c - c = 9] be said to be the same as >> Statement Number Two {9c = 9} ? are clearly not the same). Equally 11c-2c=9 is equivalent and any equation of this form which has the > solution of c=1 - of which there are an infinite number. Having just joined this ng, it wasn't clear that this was a continuation of a discussion. I would suggest that Re: or Continuation be used to indicate this fact. Far be it from me to wade in in the middle of something. Nick === Subject: Re: Man gets 8 years in prison for killing a cat <12itl4hrg7iaoa8@corp.supernews> <1615978.AL714FIcRM@example> <4906522.srnRM5uP8E@example> No, they are not. Using units I get the simpler: > (6 + sqrt(13) + 4.sqrt(37) + sqrt(13.37))/2 But they are nearly. Define: v1 = (3 + sqrt(13))/2 u1 = 6 + sqrt(37) u2 = 6 - sqrt(37) and note that they are primitive units in their quadratic field (and I think also primitive in the bi-quadratic field). Further define: p1 = (111 + 30.sqrt(13) - 18.sqrt(37) - 5.sqrt(13.37))/2 p2 = (111 + 30.sqrt(13) + 18.sqrt(37) + 5.sqrt(13.37))/2 (with defining polynomial z^4 - 222.z^3 + 625.z^2 - 444.z + 4) and q1 = (5 + 3*sqrt(13) - sqrt(37) + sqrt(13)*sqrt(37))/4 q2 = (5 + 3*sqrt(13) + sqrt(37) - sqrt(13)*sqrt(37))/4 q3 = (5 - 3*sqrt(13) - sqrt(37) - sqrt(13)*sqrt(37))/4 q4 = (5 - 3*sqrt(13) + sqrt(37) + sqrt(13)*sqrt(37))/4 (with defining polynomial z^4 - 5.z^3 - 70.z^2 + 371.z + 3). We find: p1.p2 = 2; as Keith predicted (my p1 and p2 are the same as Keith's upto units), and p1.p2.q1.q2.(-v1^2) = 5 - sqrt(13) q1.q3.u2 = (5 + sqrt(37))/2 q2.q4.u1 = (5 - sqrt(37))/2 p1.q1.q4.u1 = (sqrt(37 + sqrt(13))/2 p2.q2.q3.(-u2) = (sqrt(37) - sqrt(13))/2 p1.q1^2.(v1^2) = 5 + (sqrt(37) - sqrt(13))/2 p2.q2^2.(v1^2) = 5 - (sqrt(37) + sqrt(13))/2 just as Rupert predicted (and note that the units play not a role in factorisation and co-primeness). (And this all due to Maple and some probing.) This is in a form even JSH should be able to understand. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: probability problem > > >You don't have to sum any series (assuming you've figured out the >distribution of the number of throws and you remember its mean). > >It's just a simple geometric series, so I didn't see the need for >>anything fancier. >> >What >theorem do you know about > E(X_1 + X_2 +...+X_N) >where the X's are iid and N is stopping time? > >I don't know anything about that. I just used the theorem >>E(X_1 + X_2 + ...) = E(X_1) + E(X_2) + .... >> Then you got the right answer for the wrong reason. The fact is that S >= X_1 + ...+ X_N is a sum of a random number of terms, because N is >itself random. IF N were independent of the X_i you could apply a >simple probability argument to find that E(S) = E(X)*E(N), which is >essentially what you did. However, N is NOT independent of the X_i, >since whether or not we have N = n depends on what we observe about >X_n. Nevertheless, because N is a so-called STOPPING TIME, the previous >result goes through, but this is a reasonably deep theorem, usually >not discussed in introductory courses, for example. Look up the topic >of Renewal Theory to see more of this stuff. > > I, too, would apply Wald's Theorem, which is not covered in a first course in probability. However, one *can* do this with a geometic random variable or a geometric series. Note that the number of tosses *before* the first one or six has geometric distribution starting at 0 with parameter 1/3. Each of the tosses are uniformly dsitributed on {2, 3, 4, 5}. The final toss is uniformly distributed on {1, 6}. >[...] > > <>Here X_n has range {0,1,2,3,4,5,6}, with P(X_n > 0) = (2/3)^(n-1) and >> the nonzero values distributed uniformly, so E(X_n) = 3.5*(2/3)^(n-1). > Actually, this also works. This is basically the proof of Wald's Theorem. Vickson's chastisement was unmerited. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: probability problem >> One throws a fair dice until he gets 1 or 6. What would >> be the expectation for the sum of all the numbers you obtained? >10.5 >> any ideas how to solve this? >Sum an infinite geometric series. The expected number on a throw is 3.5 whether or not the throw is a stopping throw or not, so the expected sum is 3.5*H, where H is the expected number of throws. This also can be computed without summing a geometric series. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: probability problem > Referring to your definition of X_n below, I cannot see anything that > prevents you from having X_n = 0 but X_(n+1) > 0, so I don't see your > process ever stopping. Also: the only way the process stops at toss n > is to have X_n = 1 or 6, so THIS X_n has a different distribution from > the others. Just _by accident_, the unconditional E(X) is the same as > the conditional E(X|X=1 or 6) and E(X|X not 1 or 6), but a change in > the description (say to a 7-sided die, or to stopping when 6 alone is > reached) would invalidate these equalities. The really great thing > about the stopping time theorem is that we can be assured that we don't > need to worry about this stuff, but it is far from obvious! This conversation is getting weird, but I'll take another stab at it. My process is the original poster's process: a die is rolled until it comes up 1 or 6; the random variable X is the sum of all the numbers rolled including the last one, which of course has to be a 1 or 6; we want E(X). (The particular numbers 1 and 6 are irrelevant; nothing changes and the answer is still 10.5 if, e.g., the game stops when a 1 or 2 comes up.) Because you mentioned it, let Y be the number that comes up on the last roll. So Y has the distribution f(1) = f(6) = 1/2. Well and good, but I'm not going to use the variable Y. I define an *infinite* sequence of random variables that I call X_n (n = 1, 2, 3, ...). Maybe this is what's bothering you? Maybe X_n is supposed to mean something else in this sort of problem, and my notation is nonstandard? In that case, feel free to change X_n to U_n or V_n or whatever letter is appropriate. X_1 is the number that comes up on the first roll; its probability (mass)distribution is f(1)=f(2)=f(3)=f(4)=f(5)=f(6)=1/6, and E(X_1) = 7/2. X_2 has the value 0 if 1 or 6 came up on the first roll, otherwise its value is the number that comes up on the second roll. Its probability distribution is f(0) = 1/3, f(1)=f(2)=f(3)=f(4)=f(5)=f(6)=1/9, and E(X_2) = 7/3. X_3 = 0 if 1 or 6 came up on the first or second roll, otherwise it's the number that comes up on the third roll. Its probability distribution is f(0) = 5/9, f(1)=f(2)=f(3)=f(4)=f(5)=f(6)=2/27; and E(X_3) = 14/9. In general (and as I said before), X_n has the value 0 if 1 or 6 came up before the nth roll, otherwise its value is the number that comes up on the nth roll. The distribution of X_n is P(X_n = 0) = 1-(2/3)^(n-1), and for k in {1,2,3,4,5,6}, P(X_n = k) = (1/6)*(2/3)^(n-1). Thus E(X_n) = 3.5*(2/3)^(n-1). Is everything OK so far? The X_n's are certainly dependent, since X_n in {0,1,6} implies X_{n+1) = 0. I don't care about that. All that matters is that the sum of the X_n's is equal to the original poster's variable X, and E(X_n) = 3.5*(2/3)^(n-1). Hence E(X) is the sum of the infinite geometric series with first term 3.5 and common ratio 2/3, i.e., E(X) = 10.5. === Subject: Re: probability problem > One throws a fair dice until he gets 1 or 6. What would > be the expectation for the sum of all the numbers you obtained? >any ideas how to solve this? The following is equivalent to some of the solution methods already presented or mentioned, but perhaps more transparent: Let g be the value of the desired expectation. At the first roll, events b and c can happen, with b being a 1 or 6, and c a 2, 3, 4, 5. g = Pr(b)*E(b) + Pr(c)*E(c). Now Pr(b)=1/3, Pr(c)=2/3, E(b)=(1+6)/2, and E(c)=(2+3+4+5)/4 + g. Solving g = (1/3)*3.5 + (2/3)*(14/4 + g) gives 3g = 3.5 + 2*(3.5+g), or g=10.5. -jiw === Subject: Re: Yau made a fool of himself. Again. > Yau has hired lawyers to attack Sylvia Nasar, what a shame. > --Chen If you look at the link www.doctoryau you will see numerous letters of support from many matheamticians have recently been posted there defending S.T.Yau's reputation. S.T.Yau has never done anything to detract from Perelman and in fact has supported the awarding of the Fields Medal to Perelman. Perelman's unfunded lifestyle has been by choice. He was offered excellent positions in the United States and turned them down before he ever began work on the Poincare Conjecture. No one ever held him back. We don't actually need more mathematicians like Perelman who refuse to publish and teach. We need mathematicians like S.T.Yau. === Subject: Closed and compact sets If A is contained in R^k and B is contained in R^k, define A + B to be the set of all sums x + y with x belonging to A and y belonging to B. If K is compact and C is closed in R^k, prove that K + C is closed. Hint: Take z not belonging to K + C, put F = z - C, the set of all z - y with y belonging to C. Then K and F are disjoint. Choose a delta. Show that the open ball with center z and radius delta does not intersect K + C. === Subject: Re: Closed and compact sets While I make it a policy not to give away complete answers on the web, I recommend you try using the accumulation points definition of closed and sequences to prove this statement. It is much easier then using open sets. === Subject: Re: New Yorker on the Poincare Conjecture >We asked [Perleman] whether he had read Cao and Zhu.92s >paper. 'It is not clear to me what new contribution did > they make,' he said. 'Apparently, Zhu did not quite > understand the argument and reworked it.' It does not say here that Perelman read the paper. Ordinarily a mathematician will read a paper that concerns his work and can specifically address why he has different hypothesis. Perelman chose to leave mathematicians out in the cold for over a year not responding to their questions about his work, so it isn't suprising that people had to rework the arguement to make it rigorous. Let me clarify one point: before the Cao-Zhu paper came out no one was claiming Perelman had succeeded in proving the Geometrization Conjecture. Morgan and Tian only check Poincare (which is significantly less than Geometrization) and even they need 300 pages to clarify Perelman's work. Morgan and Tian are admirable in giving all the credit to Perelman although they are world class mathematicians and spent significant time working out the details themselves. Cao-Zhu spent the extra time to work out geometrization and felt no obligation to try to imitate Perelman's work to the letter when it was easier for them to adapt some of the lemmas. While Yau has given them significant credit for this crowning achievement (eg culmination), he has not actually claimed they deserve such a high percent of credit as was published by the Chinese media and quoted in the New Yorker. He clarified this to Nasar when she interviewed him. === Subject: Re: Nonlinear equations You gave a convincing solutions of the equations Professor Israel. For my own understanding I will comment upon the following. Since the RHS of (1) is irrational the LHS must also be irrational of the same form. Since in (1) x has odd powers and y has even power the choice x = sqrt(X) and y = sqrt(Y) seem to be logical. But you have chosen(correctly) x = Re(... ... )^1/3 and y = Re( ... ... )^1/3 Your comments will be highly appreciated. >Consider the following three equations: >x^3 - 3xy^2 = sqrt(A) (1) >3x^2y - h^3 = sqrt(B) (2) I assume you meant 3 x^2 y - y^3 = sqrt(B), since h appears > nowhere else in the question. >x^2 + y^2 = C (3) >A, B, C are integers each > 1 and none is a pefect square. >Assertion: The three equations can be satisfied only if x = sqrt(X) >and y = sqrt(Y) >where both X, Y are integers but not perfect squares. >Any comment upon the correctness of the assertion will be appreciated. sqrt(A) + i sqrt(B) = (x+iy)^3 > sqrt(A) - i sqrt(B) = (x-iy)^3 > C = x^2 + y^2 = (A+B)^(1/3) Counterexample: > Take C = 2, A = 3, B = 5 (so C^3 = A + B) > x = Re((sqrt(3) + i sqrt(5))^(1/3)) > y = Im((sqrt(3) + i sqrt(5))^(1/3)) > x^2 and y^2 are not integers. > In fact x is algebraic of degree 6, a root of the polynomial > 16*z^6-48*z^4+36*z^2-3. Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada === Subject: Re: Mathematics Puzzle...help! > I am trying to figure out a puzzle my friend gave me the other day, but > I'm not finding any luck. If anyone could help, I would really > appreciate it. > Find a 10 digit number using each of the numbers from 0 to 9 once such > that the last i digits are perfectly divisible by i. Ex. 120; 0/1=0, > 20/2=10, 120/3=4, etc. > Appreciate it. I've managed to narrow it down by eliminating 98% of > the possible 2.7 million possibilities. Now I've reached a point of > trial and error--but I'm facing 55,440 trials, so any help would be The key is the seven-digit number. Not only must it be divisible by 7, > but it must also be divisible by 8 and by 10 . When an appropriate > 7-digit number is found, the puzzle is solved! I don't know how many numbers between 1,234,560 and 8,765,430 > are divisible by 560 and have seven different digits; but there are > only 20,160 permutations of 8 numbers taken 6 at a time! On > the average, you should expect about 20,160 / 56 ~ 360 to > qualify. The discussion above is based on the fact that the 10th digit must > be 9 and the first must be 0. There are other considerations > that significantly reduce the number of trials to much less than 360! > Bill J .Actually, the numbers from the 3-digit number thru the 10.digit number must all be divisible by 8. The reason is simple.. Why? Anybody? Bill J === Subject: Re: Generalized factoring idea > Factoring may be simpler than previously believed as consider the > following generalized factoring idea: > Consider x^2 - y^2 = 0 mod T and k^2 = 2xk mod T where all are integers, then it's trivial to solve out to find (x+k)^2 = y^2 + 2k^2 + nT where I introduce n as you have to pick it as well as k. One simple idea is to let n=-1, and then find the smallest k such that 2k^2 - T is positive which will give a smaller number to factor as you factor 2k^2 + nT to find y and then x. I had earlier ideas in this area where I also used a variable S, but realized that approach was unlikely to work for simple reasons. ___JSH === Subject: Re: Generalized factoring idea > Factoring may be simpler than previously believed as consider the > following generalized factoring idea: > Consider x^2 - y^2 = 0 mod T and k^2 = 2xk mod T where all are integers, then it's trivial to solve out to find (x+k)^2 = y^2 + 2k^2 + nT where I introduce n as you have to pick it as well as k. One simple idea is to let n=-1, and then find the smallest k such that 2k^2 - T is positive which will give a smaller number to factor as you factor 2k^2 + nT to find y and then x. I had earlier ideas in this area where I also used a variable S, but realized that approach was unlikely to work for simple reasons. ___JSH === Subject: Re: JSH: Generalized factoring idea [added JSH: to subject] [jstevh@msn] > Consider x^2 - y^2 = 0 mod T and k^2 = 2xk mod T where all are integers, then it's trivial to solve out to find (x+k)^2 = y^2 + 2k^2 + nT where I introduce n as you have to pick it as well as k. One simple idea is to let n=-1, and then find the smallest k such that 2k^2 - T is positive which will give a smaller number to factor as you factor > 2k^2 + nT to find y and then x. So you pick n, solve for a particular k via minimizing 2k^2 - T > 0, and then solve (x+k)^2 = y^2 + 2k^2 + nT [1] for pairs. You realize you haven't /used/ your k^2 = 2xk mod T [2] equation at all then? Alas, there's no reason to hope that [1] will lead to a non-trivial factor of T, except by blind luck, unless [2] is satisfied by an x you happen to get by solving [1]. But just like last time around, you have no way to force [2] to hold, and most x obtained by solving [1] won't satsify [2]. > I had earlier ideas in this area where I also used a variable S, but > realized that approach was unlikely to work for simple reasons. And, as above, a correct simple reason for poor performance still applies to this version. If you can't understand this, try small numeric examples. The problem is obvious. One way to repair that is to set k=0. Then equation [2] is trivially satisfied regardless of x, and [1] reduces to something you can read about in books. === Subject: Re: JSH: Generalized factoring idea Consider > x^2 - y^2 = 0 mod T > and > k^2 = 2xk mod T > where all are integers, then it's trivial to solve out to find > (x+k)^2 = y^2 + 2k^2 + nT > where I introduce n as you have to pick it as well as k. > One simple idea is to let n=-1, and then find the smallest k such that > 2k^2 - T > is positive which will give a smaller number to factor as you factor > 2k^2 + nT to find y and then x. So you pick n, solve for a particular k via minimizing 2k^2 - T > 0, and > then solve (x+k)^2 = y^2 + 2k^2 + nT [1] for pairs. You realize you haven't /used/ your k^2 = 2xk mod T [2] equation at all then? Alas, there's no reason to hope that [1] will lead to > a non-trivial factor of T, except by blind luck, unless [2] is satisfied by > an x you happen to get by solving [1]. But just like last time around, you > have no way to force [2] to hold, and most x obtained by solving [1] won't > satsify [2]. > You are just being stupid. Try it. I tested a few examples. The math has to work with what you give it, and it may choose T=1, but otherwise it has to pick some piece of what you select. I guess you're just desperate at this point, but lying is just dumb. It also proves you are a con, who can't figure out when to start telling the truth. Try the idea. Put up some examples that prove your point if I'm wrong. The math IS EASY. === Subject: Re: JSH: Generalized factoring idea [jstevh@msn] > Consider >> x^2 - y^2 = 0 mod T >> and >> k^2 = 2xk mod T >> where all are integers, then it's trivial to solve out to find >> (x+k)^2 = y^2 + 2k^2 + nT >> where I introduce n as you have to pick it as well as k. >> One simple idea is to let n=-1, and then find the smallest k such that >> 2k^2 - T >> is positive which will give a smaller number to factor as you factor > 2k^2 + nT to find y and then x. [Tim Peters] >> So you pick n, solve for a particular k via minimizing 2k^2 - T > 0, and >> then solve >> (x+k)^2 = y^2 + 2k^2 + nT [1] >> for pairs. You realize you haven't /used/ your >> k^2 = 2xk mod T [2] >> equation at all then? Alas, there's no reason to hope that [1] will lead >> to a non-trivial factor of T, except by blind luck, unless [2] is >> satisfied by an x you happen to get by solving [1]. But just like last >> time around, you have no way to force [2] to hold, and most x obtained by >> solving [1] won't satsify [2]. [jstevh@msn] > You are just being stupid. Try it. I did, and, as I said, the problem is /obvious/ if you have a functioning brain cell. You ignored my mathetical points entirely, suggesting that you don't. > I tested a few examples. Then show your work. It's always possible that I'm missing your meaning. An example would clarify that. Here's one off the top of my head: T = 8383 = 83 * 101 pick n = -1 minimize 2k^2 - 8383 > 0 to get k = 65 (at k=64 it's negative) check: 2*65^2 - T = 67, which happens to be prime There are only four ways to factor 67 as a product of two integers, and your methods always suffer then because they're essentially pseudo-random number generators (you need a very smooth surrogate to generate tons of gcd candidates, and, tough luck, but 67 generates very few): 1 * 67 67 * 1 -1 * -67 -67 * -1 For each way, f1*f2, [1] gives (x+k)^2 - y^2 = (x+k+y)*(x+k-y) = f1 * f2 = 2k^2 - T = 67 so: x+y = f1-k x-y = f2-k follow. This gives 4*2 = 8 gcds (with T) to try, but only 4 are distinct: +/-1 - 65 +/-67 - 65 None of them reveal a non-trivial factor; the gcd is 1 for each. This is typical, as I've tried far more than a few examples -- and, as I said, the problem is obvious if you have a functioning brain cell. > The math has to work with what you give it, and it may choose T=1, but > otherwise it has to pick some piece of what you select. I guess you're just desperate at this point, Nope. What on Earth would I be desperate /about/? > but lying is just dumb. It also proves you are a con, who can't figure > out when to start telling the truth. Uh huh. > Try the idea. Put up some examples that prove your point if I'm wrong. Your turn to put up an example that works. I've generated millions that don't work. > The math IS EASY. And it's indeed very easy to see why it works only by luck. Again, in the absence of a way to /force/ [2] to be true, equation [1] leads to a factor only by luck. If [2] /is/ true, then [1] implies the difference of squares is congruent to 0 modulo T, and then there's some sense to this. But when [2] isn't true, [1] is crap. You still have no way to force [2]. === Subject: Re: JSH: Generalized factoring idea Consider >> x^2 - y^2 = 0 mod T >> and >> k^2 = 2xk mod T >> where all are integers, then it's trivial to solve out to find >> (x+k)^2 = y^2 + 2k^2 + nT >> where I introduce n as you have to pick it as well as k. >> One simple idea is to let n=-1, and then find the smallest k such that >> 2k^2 - T >> is positive which will give a smaller number to factor as you factor > 2k^2 + nT to find y and then x. [Tim Peters] >> So you pick n, solve for a particular k via minimizing 2k^2 - T > 0, and >> then solve >> (x+k)^2 = y^2 + 2k^2 + nT [1] >> for pairs. You realize you haven't /used/ your >> k^2 = 2xk mod T [2] >> equation at all then? Alas, there's no reason to hope that [1] will lead >> to a non-trivial factor of T, except by blind luck, unless [2] is >> satisfied by an x you happen to get by solving [1]. But just like last >> time around, you have no way to force [2] to hold, and most x obtained by >> solving [1] won't satsify [2]. [jstevh@msn] > You are just being stupid. Try it. I did, and, as I said, the problem is /obvious/ if you have a functioning > brain cell. You ignored my mathetical points entirely, suggesting that you > don't. > I tested a few examples. Then show your work. It's always possible that I'm missing your meaning. > An example would clarify that. Here's one off the top of my head: T = 8383 = 83 * 101 > pick n = -1 > minimize 2k^2 - 8383 > 0 to get k = 65 (at k=64 it's negative) > check: 2*65^2 - T = 67, which happens to be prime There are only four ways to factor 67 as a product of two integers, and your > methods always suffer then because they're essentially pseudo-random > number generators (you need a very smooth surrogate to generate tons of > gcd candidates, and, tough luck, but 67 generates very few): 1 * 67 > 67 * 1 > -1 * -67 > -67 * -1 For each way, f1*f2, [1] gives (x+k)^2 - y^2 = (x+k+y)*(x+k-y) = f1 * f2 = 2k^2 - T = 67 so: x+y = f1-k > x-y = f2-k follow. This gives 4*2 = 8 gcds (with T) to try, but only 4 are distinct: +/-1 - 65 > +/-67 - 65 None of them reveal a non-trivial factor; the gcd is 1 for each. This is > typical, as I've tried far more than a few examples -- and, as I said, > the problem is obvious if you have a functioning brain cell. > Show k^2 - 2xk to see what the math thinks T is. > The math has to work with what you give it, and it may choose T=1, but > otherwise it has to pick some piece of what you select. > I guess you're just desperate at this point, Nope. What on Earth would I be desperate /about/? > You've lost. Show what the math thinks T is, by showing k^2 - 2xk, and comparing with x^2 - y^2, and you'll see how it works. I didn't say the math had to factor your choice of T the way you want it to, as it can choose for T=1, or the full composite without factoring it. Your problem is you need to convince people against the truth, while I can show that you're doing it. === Subject: Re: JSH: Generalized factoring idea [added JSH: to subject] > [jstevh@msn] >Consider > x^2 - y^2 = 0 mod T > and > k^2 = 2xk mod T > where all are integers, then it's trivial to solve out to find > (x+k)^2 = y^2 + 2k^2 + nT > where I introduce n as you have to pick it as well as k. > One simple idea is to let n=-1, and then find the smallest k such that > 2k^2 - T > is positive which will give a smaller number to factor as you factor >2k^2 + nT to find y and then x. > So you pick n, solve for a particular k via minimizing 2k^2 - T > 0, and > then solve > (x+k)^2 = y^2 + 2k^2 + nT [1] > for pairs. You realize you haven't /used/ your > k^2 = 2xk mod T [2] > equation at all then? Alas, there's no reason to hope that [1] will lead to > a non-trivial factor of T, except by blind luck, unless [2] is satisfied by > an x you happen to get by solving [1]. But just like last time around, you > have no way to force [2] to hold, and most x obtained by solving [1] won't > satsify [2]. >You are just being stupid. Try it. I tested a few examples. The math has to work with what you give it, and it may choose T=1, but > otherwise it has to pick some piece of what you select. I guess you're just desperate at this point, but lying is just dumb. > It also proves you are a con, who can't figure out when to start > telling the truth. Try the idea. Put up some examples that prove your point if I'm wrong. The math IS EASY. > How about you factor a large number with your approach. If it is so easy, prove it. I know you won't actually do it. You are bluffing as usual. You are afraid as usual to actually try something, because you know it will be another dead end. === Subject: Re: Generalized factoring idea > Factoring may be simpler than previously believed as consider the > following generalized factoring idea: With T the target composite: x^2 - y^2 = 0 mod T is how far researchers previously went, and that area is well-worked > showing it difficult to factor T as T increases in size, but my > research shows it to just be a primitive case of a more general > solution found by using two additional variables, S and k, where S - 2xk = 0 mod T which allows you to now use quadratic methods as usual as you easily > then have (x+k)^2 = y^2 + S + k^2 + nT where n is some non-zero integer, and notice, importantly, these > generalized factoring equations default to the well-known ones with > S=k=0. But with S and k non-zero they indicate a factorization of S + k^2 + nT > as the route to factoring T itself, as the general factoring method. My previous postings in this area from a quirk of how I did the > research used n=0, and that not surprisingly didn't work! The math > doesn't know what T is if n=0, so you get random behavior, and > objections raised against these ideas depend on that quirk of how I > initially talked about it. They have to as the idea includes previously known factoring methods, > and to seem to fail there had to be a simple reason. However, with n nonzero, thorough analysis of when the ideas shown here > lead to a non-trivial factorization of a composite T do not show the > normal rules, like indications that the size of T matters. I've just > done a bit of analysis in this area and as of yet have found no > indication that these ideas cannot be made practical, though I haven't > done it myself, only having done initial theory. But consider, all that I did in actuality was find a more generalized > set of factoring equations, which include those typically used in > previously known approaches when S=k=0. No demonstrations to date have been given that this approach fails as > previous arguments on Usenet centered on cases where for reasons having > to do with how I derived the equations, initially I focused on n=0, > disconnecting T itself in a key place, giving random behavior. > What happened to the coup de grace? Did I miss the press conference? Marcus. === Subject: Re: JSH: Generalized factoring idea [added JSH: to subject; dropped sci.crypt] [jstevh@msn] >> Factoring may be simpler than previously believed as consider the >> following generalized factoring idea: >> ... [marcus_b] > What happened to the coup de grace? Did I miss the press conference? Don't complain -- at least he skipped the Death to the Infidels phase of the cycle this time (or kept it to himself), and that's a good thing even for him. All his blog entries related to the coup de grace vanished several days ago, and in the last 24 hours his AboutMyMath Google Group vanished too (although his blog still links to it). An emergency meeting of the JSH Response Team convened last night, to decide on the official end time of cycle #46, and the official start time of cycle #47. Unfortunately, when you get 30 top mathematicians together in the same room, they waste most of their time trying to get their public story straight. It's worse than usual this time, as Pollard has so far been unable to get Witten to understand what factoring /is/ -- it's like N is just too small a set for him to conceive of. So, despite that it sure looks like cycle 47 started two days ago, /officially/ we're still in 46. Top mathematicians working undercover here are urged not to respond before the Team distributes model replies for the new cycle. === Subject: Re: JSH: Generalized factoring idea > Factoring may be simpler than previously believed as consider the >> following generalized factoring idea: >> ... [marcus_b] > What happened to the coup de grace? Did I miss the press conference? Don't complain -- at least he skipped the Death to the Infidels phase of > the cycle this time (or kept it to himself), and that's a good thing even > for him. All his blog entries related to the coup de grace vanished several days > ago, and in the last 24 hours his AboutMyMath Google Group vanished too > (although his blog still links to it). An emergency meeting of the JSH Response Team convened last night, to decide > on the official end time of cycle #46, and the official start time of cycle > #47. Unfortunately, when you get 30 top mathematicians together in the same > room, they waste most of their time trying to get their public story > straight. It's worse than usual this time, as Pollard has so far been > unable to get Witten to understand what factoring /is/ -- it's like N is > just too small a set for him to conceive of. So, despite that it sure looks like cycle 47 started two days ago, > /officially/ we're still in 46. Top mathematicians working undercover here > are urged not to respond before the Team distributes model replies for the > new cycle. An addendum. The unravelling of Establishment Math, i.e., Galois Theory As It Is Now Taught, was JSH's astonishing discovery this week that 49 - 5 = 42. Yup, it's that simple, people. You have been conned by schoolmarms for centuries. Now you're going to have to face some tough questioning by angry mobs. Marcus. === Subject: Re: JSH: Generalized factoring idea Fellow conspirators, ... below. > [added JSH: to subject; dropped sci.crypt] [jstevh@msn] >> Factoring may be simpler than previously believed as consider the >> following generalized factoring idea: >> ... [marcus_b] > What happened to the coup de grace? Did I miss the press conference? Don't complain -- at least he skipped the Death to the Infidels phase of > the cycle this time (or kept it to himself), and that's a good thing even > for him. > Actually I was not sure whether it was coup de grace or coup d'etat - both seemed equally likely, what with the collapse of the world economic system, military takeover of all math departments (these guys have been waiting for an excuse to do this for YEARS), cracking (a la Paris Hilton) of the cell phones of Brittany Spears, Madonna, Tony LaRussa, Donald Rumsfeld, James K. Polk, Mr. Whipple, and Mike Tyson, and infiltration of the NSA by Surrogate Factoring groupies. > All his blog entries related to the coup de grace vanished several days > ago, and in the last 24 hours his AboutMyMath Google Group vanished too > (although his blog still links to it). My God. The Apocalypse cannot be far off. An emergency meeting of the JSH Response Team convened last night, to decide > on the official end time of cycle #46, and the official start time of cycle > #47. Unfortunately, when you get 30 top mathematicians together in the same > room, they waste most of their time trying to get their public story > straight. It's worse than usual this time, as Pollard has so far been > unable to get Witten to understand what factoring /is/ -- it's like N is > just too small a set for him to conceive of. > Jesus. Why weren't we told? I bet 'Lapdog' Ullrich is behind this. Him and his henchmen. Magidin, Rupert, Hughes, Baron, Winter, the usual suspects. Yes, round 'em up. Might as well bring in Wiles, Ribet, Mazur, Granville, Langlands, Odlyzko, Perelman, Rivest-Shamir-Adelman, Grothendieck, Bourbaki, and the others as well. Stand them all up in a straight line and just use one bullet. This is Extreme Math, baby! > So, despite that it sure looks like cycle 47 started two days ago, > /officially/ we're still in 46. Top mathematicians working undercover here > are urged not to respond before the Team distributes model replies for the > new cycle. Time to go underground, I'd say. The lurking rabble are demand- ing our hides. JSH has stirred them from their apathy. Once they get a whiff of Real Math, they just want raw meat. Get back, rabble! Ah, for the days when they would accept whatever bull pablum we would put out - Dedekind, Hilbert, Minkowski, Godel, Cantor, that crowd of hacks. They served us well at least for a while. They barely understood the quadratic formula, but slavish sheeplike grad students bought it for decades. Now we are driven out of our welfare Paradise. Curse you, JSH! Marcus. === Subject: Re: JSH: Generalized factoring idea > Factoring may be simpler than previously believed as consider the >> following generalized factoring idea: >> ... [marcus_b] > What happened to the coup de grace? Did I miss the press conference? Don't complain -- at least he skipped the Death to the Infidels phase of > the cycle this time (or kept it to himself), and that's a good thing even > for him. All his blog entries related to the coup de grace vanished several days > ago, and in the last 24 hours his AboutMyMath Google Group vanished too So that title is now available? > (although his blog still links to it). An emergency meeting of the JSH Response Team convened last night, to decide > on the official end time of cycle #46, and the official start time of cycle > #47. Unfortunately, when you get 30 top mathematicians together in the same > room, they waste most of their time trying to get their public story > straight. It's worse than usual this time, as Pollard has so far been > unable to get Witten to understand what factoring /is/ -- it's like N is > just too small a set for him to conceive of. So, despite that it sure looks like cycle 47 started two days ago, > /officially/ we're still in 46. Top mathematicians working undercover here > are urged not to respond before the Team distributes model replies for the > new cycle. === Subject: Re: JSH: Generalized factoring idea [Tim Peters] >> ... >> All his blog entries related to the coup de grace vanished several days >> ago, and in the last 24 hours his AboutMyMath Google Group vanished too [mensanator@aol] > So that title is now available? I suppose you could try it. The help files say: ... Please note that after a Google Group is created, even if it's later deleted, its name is stored in our index and becomes unavailable for future use by other groups. ... The group in question was: This afternoon I happened to notice it had vanished from my list of recently visited groups, and I know it was still there last night. === Subject: Re: Generalized factoring idea > It is completely worthless drivel, with no useful ideas, old recycled JSH > crap, complete bogus, scratching in the dirt, broken logic, a single brown > shoe. I ressemble that remark! My father, who lately had only one leg, found a single brown shoe useful. === Subject: Re: Generalized factoring idea <4540c8cd$0$97245$892e7fe2@authen.yellow.readfreenews.net> It is completely worthless drivel, with no useful ideas, old recycled JSH > crap, complete bogus, scratching in the dirt, broken logic, a single brown > shoe. I ressemble that remark! My father, who lately had only one leg, found > a single brown shoe useful. But it would have to be for the correct foot, eh? === Subject: Re: Generalized factoring idea > >It is completely worthless drivel, with no useful ideas, old recycled JSH >crap, complete bogus, scratching in the dirt, broken logic, a single brown >shoe. >>I ressemble that remark! My father, who lately had only one leg, found >>a single brown shoe useful. But it would have to be for the correct foot, eh? > OK, a 50% chance of being useful. === Subject: Re: Generalized factoring idea editing attributions>. What if? Any idea how that scenario could be blocked? >>By not replying to idiots who post about random factoring ideas? Assumptions: > There is ample evidence, that a posting advising David Ullrich to > stop replying to JSH will have no effect. > It is well known that David Ullrich replies to JSH posts. You might make an addendum: Ullrich replies to JSH much less often than many others do. So why does TCD single him out? There seems to be something personal here. === Subject: Re: Generalized factoring idea > >>Well, sure, but who bothers with the math parts? Except for a little >>research in Mathworld and wikipedia to educate myself on the new terms >>James brings in, I just read it for the entertainment value. You should consider getting cable and watching wrestling ... >At least that's funny. >JSH is just annoying because it's the same crap accusations and lame > math over and over. JSH's contributions are _much_ funnier than yours. === Subject: Re: I found a way to get Surrogate Factoring to work!; was Re: JSH: Generalized factoring idea [Proginoskes] > I think he kill-files me. A megalomaniac missing a chance to see his name? Not likely -- and remember that this is the fellow who measures the impact of his research by ego-surfing. Besides, he appears to use Google Groups for Usenet, and it doesn't offer kill files. > Nevertheless ... ((A few minutes later...)) I found a way to get Surrogate Factoring to work! Factoring in Linear > Time! You need some high-power math tools to do it (Generalized Factored > Y-Invariants), but once you get the concept, it's easy. directly to the coverage problem of the algebraic integers too. > I've encoded it and posted the algorithm at > http://www.amishrakefight.org/gfy/ . Get it before the NSA does! Yup, clearly on the right track. So far I'm missing the connection with watermelons, but GFYs are still new to me. === Subject: Re: Generalized factoring idea On 26 Oct 2006 08:16:57 -0700, pomerado@hotmail > Well, sure, but who bothers with the math parts? Except for a little >> research in Mathworld and wikipedia to educate myself on the new terms >> James brings in, I just read it for the entertainment value. >> You should consider getting cable and watching wrestling ... >> At least that's funny. >> JSH is just annoying because it's the same crap accusations and lame >> math over and over. I gather you don't like him, and you think the threads he atarts are a >waste of time. But you still read them? You're jumping to conclusions. He knows things about what I think that I didn't know - given that there's no reason to suppose he has to read a thread to know what it contains. David C. Ullrich === Subject: Re: Generalized factoring idea > Factoring may be simpler than previously believed as consider the > following generalized factoring idea: With T the target composite: x^2 - y^2 = 0 mod T is how far researchers previously went, and that area is well-worked > showing it difficult to factor T as T increases in size, but my > research shows it to just be a primitive case of a more general > solution found by using two additional variables, S and k, where S - 2xk = 0 mod T which allows you to now use quadratic methods as usual as you easily > then have (x+k)^2 = y^2 + S + k^2 + nT where n is some non-zero integer, and notice, importantly, these > generalized factoring equations default to the well-known ones with > S=k=0. But with S and k non-zero they indicate a factorization of S + k^2 + nT > as the route to factoring T itself, as the general factoring method. My previous postings in this area from a quirk of how I did the > research used n=0, and that not surprisingly didn't work! The math > doesn't know what T is if n=0, so you get random behavior, and > objections raised against these ideas depend on that quirk of how I > initially talked about it. They have to as the idea includes previously known factoring methods, > and to seem to fail there had to be a simple reason. However, with n nonzero, thorough analysis of when the ideas shown here > lead to a non-trivial factorization of a composite T do not show the > normal rules, like indications that the size of T matters. I've just > done a bit of analysis in this area and as of yet have found no > indication that these ideas cannot be made practical, though I haven't > done it myself, only having done initial theory. But consider, all that I did in actuality was find a more generalized > set of factoring equations, which include those typically used in > previously known approaches when S=k=0. No demonstrations to date have been given that this approach fails as > previous arguments on Usenet centered on cases where for reasons having > to do with how I derived the equations, initially I focused on n=0, > disconnecting T itself in a key place, giving random behavior. Do you have a job, or do you just do this all day? === Subject: question with ms-word or powerpoint. hello sir~ http://board-2.blueweb.co.kr/user/math565/data/math/eq.jpg i want to find the inequality symbol <= in ms-programs. i can find the inequality symbol <- in ms-programs. but i can't find the inequality symbol <= can you find this symbol ? === Subject: Re: Growth Rate of Level-k Goodstein Function <45363492$0$24622$b45e6eb0@senator-bedfellow.mit.edu> Okay, I'm going to make an attempt to answer CH's question about the Howard ordinal. First, I will describe a standard notation for the Howard ordinal. An ordinal a is additively principal if for all ordinals b,c less than a, b + c is less than a. The additively principal ordinals are exactly those of the form w^a for some a. Cantor's Theorem says that each ordinal can be written uniquely as a weakly decreasing sum of additively principal ordinals. So every ordinal can be written uniquely as a = Sum (i = 1 to k) w^a_i, (i < j -> a_i >= a_j) which is known as Cantor normal form. By recursively applying the above notation, we eventually reduce all ordinals to either 0 or epsilon numbers. For our Howard ordinal notation, we use a function theta that represents all epsilon numbers. Theta will be a two variable function. The second variable will be also be less than the Howard ordinal, so it will be recursively represented using our notation; however, the first variable can in fact be greater than our original ordinal! The first variable will be represented as an exponential polynomial in Omega (we will just write O), where Omega is typically identified with a very large ordinal, usually the first uncountable ordinal. So, if we define Ordinal to be the set of ordinals up to the Howard ordinal, then we define UncountableOrdinal to be the set of ordinals a of the form a = O^a_1 * a_1 + ... + O^a_n * b_n where the a_i are in UncountableOrdinal, and the b_i are in Ordinal. Then theta is a function that takes an UncountableOrdinal and an Ordinal and maps them to an Ordinal. Now, to implement an ordinal hierarchy, you take the input ordinal a (and the input natural n) and determine whether the ordinal is 0, a successor, or limit. If the ordinal is 0 (which is easy to determine), we apply the base rule, like H_0(n) = n or F_0(n) = n+1. If the ordinal is a successor, then we apply recursion on the predecessor ordinal of a, either H_(a+1) (n) = H_a (n+1) or F_(a+1) (n) = F_a^n (n). If the ordinal is a limit ordinal, then we apply recursion again, replacing a with the nth ordinal in the fundamental sequence of a. A caveat here - fundamental sequences are not uniquely defined for all ordinals. They are pretty standard below epsilon_0, as there is pretty much a standard notation for such ordinals (Cantor Normal Form applied recursively) and the fundamental sequences are then obvious. But as you go higher and higher, there are more and more arbitrary choices to be made. The idea is that, for natural choices of fundamental sequences, the growth rate of these hierarchies will be the same. This doesn't seem like something we can prove or necessarily even make sense of for all recursive ordinals, but it seems to be true at the low levels we are dealing with. Anyway, keep in mind that the fundamental sequences that I am talking about are simply somewhat arbitrary choices, and not uniquely defined entities. In our notation, an ordinal is written as a sum of ordinals; to determine whether it is successor or limit, check the last summand. If it is 1, it is a successor, otherwise it is limit. So everything is rather straightforward, except for determining what the fundemental sequences of limit ordinals are. So that's For an ordinal a, let a[n] be the nth member of the fundamental sequence. Then, for a = a_1 + a_2 + ... + a_k, a[n] = a_1 + ... + a_k[n]. If a_k = w^(b+1), then a_k [n] = w^b + w^b + ... + w^b (n times). If a_k = w^b with b limit, then a_k[n] = w^(b[n]). So we have reduced the problem to finding the fundamental sequences of theta values. First, we have theta (a, b) [n] = theta (a, b[n]) when b is a limit ordinal. theta (a+1, 0) [0] = 1 theta (a+1, 0) [n+1] = theta (a, theta (a+1, 0) [n]) theta (a+1, b+1) [0] = theta (a+1, b) + 1 theta (a+1, b+1) [n+1] = theta (a, theta (a+1, b+1) [n]) For the remainder, let a = O^a_1 b_1 + ... + O^a_k b_k. Let a' = O^a_1 b_1 + ... + O^a_(k-1) b_(k-1). Assume a_k = 0, b_k limit. Then theta (a, 0) [0] = 1 theta (a, 0) [n+1] = theta (a[n+1], theta (a, 0) [n]) theta (a, c+1) [0] = theta (a, c) + 1 theta (a, c+1) [n+1] = theta (a[n+1], theta (a, c+1) [n]) I'll address what happens with a_k != 0 with my next post. === Subject: counting the elements of finite sets The sets {x} and {x,x} supposedly are equivalent because each element of one is also in the other. But that doesn't seem right because, eg., suppose we have the two sets, {2} and {sol(2x=4),sol(3x=6)} where 'sol' means 'solution of the equation'. It seems clear that the second set consists of two solutions, not one element, no matter that the two solutions happen to be equal. (The second set is purposely not described as {x:2x=4,x:3x=6} because that would tend to obscure the point.) For another example, consider the set of all numbers of the form p/q where p and q are positive integers smaller than 10. Clearly there are 9*9 such numbers, but their distinct values number fewer than 81 since, for example, 2/4=4/8. But, by the wording of the set's description, there are still 81 elements. If, however, the set is described as the set of rational numbers of the form p/q etc., then there's no reason to count any elements of the same value more than once. of elements in finite sets depends, not just on the value of the elements, but also on the descriptions of the sets. === Subject: Re: counting the elements of finite sets > The sets {x} and {x,x} supposedly are equivalent > because each element of one is also in the other. But > that doesn't seem right because, eg., suppose we have > the two sets, {2} and {sol(2x=4),sol(3x=6)} where 'sol' > means 'solution of the equation'. It seems clear > that the second set consists of two solutions, not > one element, no matter that the two solutions happen to > be equal. (The second set is purposely not described > as {x:2x=4,x:3x=6} because that would tend to obscure > the point.) What does sol mean? If you want to keep track of how many equations you are solving, then do so. > For another example, consider the set of all numbers > of the form p/q where p and q are positive integers > smaller than 10. Clearly there are 9*9 such numbers, > but their distinct values number fewer than 81 since, > for example, 2/4=4/8. But, by the wording of the set's > description, there are still 81 elements. If, however, > the set is described as the set of rational numbers > of the form p/q etc., then there's no reason to count > any elements of the same value more than once. If you don't want 2/4 to be the same as 4/8, then make it a set of ordered pairs (p,q). > of elements in finite sets depends, not just on the > value of the elements, but also on the descriptions of > the sets. -- David Marcus === Subject: analysis with mean value theorem. hello sir~ 1. f is differentiable on [0,1]. 2. f(0) = 0. 3. there is K >0 such that |f'(x)| <= K|f(x)| for all x in [0,1]. then, f = 0 on [0,1]. ------------------------------------------- um... i think... maybe, i must use the mean value theorem. but i can't use this perfectly. let 0 < x <= 1. |f(x)|<= |f(x)/x| = |f'(x_1)| <= K|f(x_1)|, x_1 in (0, x). |f(x_1)| <= |f(x_1)/x_1| = |f'(x_2)| <= K|f(x_2)|, x_2 in (0, x_1). |f(x_2)| <= |f(x_2)/x_2| = |f'(x_3)| <= K|f(x_3)|, x_3 in (0, x_2). ......... so, i can find a sequence (x_n). (of course, x_n -> 0). so, |f(x)| <= K|f(x_1)| <= K^2|f(x_2)| <= ..... .....<= K^n|f(x_n)| but i don't know that K^n|f(x_n)| -> 0. so, i need your advice. === Subject: integrability of f(x) Can someone help me get started solving this problem: let f(x) be defined as follows: f(x) = 1, x=1, 1/2, 1/3, 1/4, 1/5, .......... f(x) = 0 otherwise show that f(x) is integrable on [0,1] and what is the value of the integral?