mm-4669 === Subject: Re: Invariants describing conjugacy classes A good reference for these questions over an algebraically closed field is Steinberg, Conjugacy Classes in Algebraic Groups, LNM 366, 1974. See especially Corollary 2, p89. As has been suggested, things become tricky when the field is not algebraically closed, at least if the group is nonsplit. === Subject: Re: primes of the form n^2+1 Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >I thought I'd read somewhere that P-NP could be shown >equivalent to a Pi_1 statement, but maybe I'm mis-remembering? What you might be remembering is that there are slight strengthenings of the statement P != NP that are Pi_1. For example, SAT is not solvable by Boolean circuits of size n^(log n). >Well this is largely a purely subjective matter, (as of now), >but I wouldn't really say that those were mathematical. >Sure, they've been fudged into a more regular-math form, >but they're pretty clearly still just codings of essentially >logical statements. You really think so? I don't find this so clear at all. In particular the graph-theoretic examples do not strike me as codings of essentially logical statements. >Exactly. The fact that large cardinal axioms themselves are >not needed, but merely their consistency, seems to say they are >essentially logical matters, not mathematical. Perhaps this is the source of the confusion. The most natural and conceptual way to prove the statements in question is by appealing directly to *combinatorial facts about large cardinals*. That is, by studying a large cardinal mathematically, you can see that it has to contain a certain nontrivial combinatorial structure, and from this you can deduce a finitary combinatorial statement. One really does *use* the structure of the large cardinal in an essential way; in this sense the large cardinal is needed. Now technically, of course, a Pi_1 statement cannot directly imply the existence of a large cardinal. But *this* seems more of a logical technicality to me. That is, the large cardinal is not necessary in the strict sense of logical implication. But morally speaking, what's at work here is the combinatorics of large cardinals rather than the structure of logical systems. For an analogy, I would argue that the natural way to prove the Paris- Harrington theorem is by assuming that infinite sets exist and to prove an infinitary Ramsey theorem. Thus I would say that infinite sets are used and even needed for the theorem, even though strictly speaking the Paris-Harrington theorem doesn't directly imply the existence of an infinite set. Anyway, it seems clear to me that Friedman's examples are mathematical and not logical. What's not so clear is whether they're natural. I don't think that they are, quite yet. But they're getting close, at least for a combinatorialist. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: primes of the form n^2+1 Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >Perhaps this is the source of the confusion. The most natural and >conceptual way to prove the statements in question is by appealing >directly to *combinatorial facts about large cardinals*. After I posted this, it occurred to me that it might be helpful to quote Friedman himself. In his paper Finite functions and the necessary use The general strategy for using large cardinals in the integers is as follows. We start with a discrete (or finite) structure X obeying certain hypotheses H. We wish to prove that a certain kind of finite configuration occurs in X, assuming that H holds. We define a suitable concept of completion in the context of arbitrary linearly ordered sets. We verify that if X has a completion with the desired kind of finite configuration, then X already has the desired kind of finite configuration. We then show, using hypotheses H, that X has completions of every well-ordered type. We now use the existence of a suitably large cardinal lambda. Using large cardinal combinatorics, we show that in any completion of order type lambgda, the desired kind of finite configuration exists. Hence the desired kind of finite configuration already exists in X. It's also worth mentioning that proving that Friedman's propositions follow from the existence of large cardinals is much easier than proving that said propositions are unprovable in ZFC (or even in ZFC + a slightly smaller large cardinal). In my opinion, if the propositions were simply secretly encoding consistency statements, then we would expect the proofs of independence to be easy. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: singular locus of a variety defined by a determinant Let F_0, F_1, ..., F_n be symmetric real matrices (with rational coefficients, though it should not matter). Consider the map g: (x_1, ..., x_n) -> det(F_0 + sum_i x_i F_i) I'm interested in the singular locus of the variety defined by g(x_1, ..., x_n)=0, that is, the points where g and all its first order derivatives are zero. Obviously, it is possible that this singular locus contains isolated points, e.g. F_0 = null 2x2 matrix F_1 = 2x2 identity matrix g(x_1) = x_1^2 But: is it possible that the singular locus considered over the complex numbers (C^n) could have dimension greater than 0, while its intersection with the reals (R^n) has isolated points? I've so far found no example where this happens but maybe I haven't looked hard enough. Any idea? http://www-verimag.imag.fr/~monniaux/ === Subject: Re: singular locus of a variety defined by a determinant > Let F_0, F_1, ..., F_n be symmetric real matrices (with rational > coefficients, though it should not matter). Consider the map > g: (x_1, ..., x_n) -> det(F_0 + sum_i x_i F_i) I'm interested in the singular locus of the variety defined by g(x_1, > ..., x_n)=0, that is, the points where g and all its first order > derivatives are zero. Obviously, it is possible that this singular locus contains isolated > points, e.g. > F_0 = null 2x2 matrix > F_1 = 2x2 identity matrix > g(x_1) = x_1^2 But: is it possible that the singular locus considered over the complex > numbers (C^n) could have dimension greater than 0, while its > intersection with the reals (R^n) has isolated points? I've so far found no example where this happens but maybe I haven't > looked hard enough. Any idea? First a simple lemma: Let A,B nxn matrices over the reals or the complex numbers with B non-singular. Define f(x) = det(A + x.B). Then f'(0) = det(B).trace(AB^-1)) [Maybe the factor (-1)^n is missing here. But that does not matter in what follows]. Define M(x1,...,xn) = F0 + x1.F1 + ... + xn.Fn. Then g = det o M. To apply the lemma the Fi's are additionally assumed to be non-singular. Then the partial derivatives of g can be calculated as g_xi(x1,...,xn) = det(Fi).trace(M(x1,...,xn).Fi^-1) . Thus the points with vanishing gradient of g (as function over both the reals or the complex numbers) is the _linear_ subspace of dimension < n defined by Ax=0 where A has the entries a_ij = trace(Fi.Fj^-1) (or the indices reversed). only one point (x1,...,xn) with vanishing gradient. (Generically should mean on the set where every Fi and the matrix A are non-singular.) In this case the singular locus has one point at most. The result of your example (n=1) easily drops out of the reasoning. BTW: We have not used the symmetry of the Fi's yet. Is there anything more known about them? -- Best wishes, J. === Subject: Re: singular locus of a variety defined by a determinant > Let F_0, F_1, ..., F_n be symmetric real matrices (with rational > coefficients, though it should not matter). Consider the map > g: (x_1, ..., x_n) -> det(F_0 + sum_i x_i F_i) I'm interested in the singular locus of the variety defined by g(x_1, > ..., x_n)=0, that is, the points where g and all its first order > derivatives are zero. If F_0=0 (the 4x4 matrix) F_1 is [1,0,0,0;0,-1,0,0;0,0,1,0;0,0,0,-1] (i.e. diag(1,-1,1,-1)) and F_2=[0,1,0,0;1,0,0,0;0,0,0,1;0,0,1,0] (i.e. a permutation matrix corresponding to (1 2) (3 4)) then I make it that g=(x^2+y^2)^2, and its singular locus in C^2 contains the line x=i*y [i^2=-1], so it has positive dimension, but the real points on the singular locus correspond to real numbers x,y with x(x^2+y^2)=y(x^2+y^2)=0 so, unless I made a slip, we had better have x=y=0. Kevin Buzzard > But: is it possible that the singular locus considered over the complex > numbers (C^n) could have dimension greater than 0, while its > intersection with the reals (R^n) has isolated points? === Subject: Re: C1 isometric embedding of the hyperbolic plane. > I attempted to make a computer graphical rendering C1 isometric > embedding of the hyperbolic plane. > I am not sure if the result really is a C1 embedding, but the pictures > are cool: http://www.xs4all.nl/~westy31/Geometry/Geometry.html#Embed A general question: Does a surface exist in the form z = f(x,y) that is an isometric C1 embedding of the hyperbolic plane? Gerard === Subject: Re: C1 isometric embedding of the hyperbolic plane. posting-account=mlT21QoAAAAzsdpimP_9tkOQjn7kW-A4 No. The hyperbolic plane cannot be isometrically embedded in R^3. This is a theorem by Hilbert. See the book Differential Geometry of Curves and Surfaces by M.P. do Carmo. > I attempted to make a computer graphical rendering C1 isometric > embedding of the hyperbolic plane. > I am not sure if the result really is a C1 embedding, but the pictures > are cool: http://www.xs4all.nl/~westy31/Geometry/Geometry.html#Embed A general question: > Does a surface exist in the form z = f(x,y) that is an isometric C1 > embedding of the hyperbolic plane? Gerard === Subject: linear algebraic variety generated by a semialgebraic closed convex set (semidefinite programming) I'm studying the semidefinite programming problem: Given real symmetric m x m matrices (well, rational matrices in practice) F_0, F_1, ..., F_n, obtain x_1, ..., x_n real coefficients such that F = F_0 + sum_i x_i F_i >= 0 (I mean, is semidefinite positive). We suppose the F_i to be linearly independent. The set S of acceptable (x_1, ...., x_n) vectors is: * closed and convex (it is the intersection of the halfspaces defined by x^t F_0 x + sum_i (x^t F_i x) x_i >= 0 for all vector x) * semialgebraic: F is semidefinite positive if and only if the coefficients of its characteristic polynomial are alternatively nonpositive and nonnegative (the determinant is nonnegative, the next one is nonpositive, etc.), so it is defined by m polynomial inequalities. I would be tremendously helped if I could compute the linear algebraic variety generated by S (I mean, the least set of the form v_0 + Vect(v_1, ..., v_d) such that S is included in it.) By computing I mean actually obtaining v_0, v_1, ..., v_d. My intuition is that if the F_i are rational, then these should also be rational if there is at least one rational point inside S. Would somebody have any idea about this that does not involve using quantifier elimination over real closed fields? I realize that this is like a shot in the dark... === Subject: Differential equation modeling distribution of primes posting-account=3IdvtgoAAAB4FiLN3bEBpHasgaRJ6CcF Hi everyone, I've come up with a simple way to derive Gauss's prime density function using probability heuristics, alone. The derivation involves setting up a nonlinear differential equation whose solution happens to agree exactly with Gauss's estimate. I don't know if I've discovered something interesting, or [more likely] old. At any rate, here's my argument: I have found a simple way to derive Gauss's estimate of the prime density function using probability heuristics. Background .. In 1792, when only 15 years old, Gauss proposed that pi(n), the prime density function pi(n)Á[Hyphen]n/(ln n). Gauss later refined his estimate to pi(n)Á[Hyphen]Li(n), where Li(n) = Integral] (1 / ln x) dx) .. I have not seen a simple derivation for this estimate, and if it exists, I am surprised why it is not more widely used in expositions on the subject of the distribution of primes. What follows is a very short argument based on a probability model. According to this model, we'll find that the distribution of primes is governed by a non-linear differential equation of the form Q'(x) = - Q(x) Q( sqrt(x) ) / x whose solution is Q(x) = 1 / 2 ln x It's probably the only solution, but then, I don't know how to prove uniqueness. (Can there be more than one solution to any given non- linear differential equation, anyway? I suspect not.) Anyway, to me, what's interesting is how using some specious probability arguments about the distribution primes I was able to set up the equation and try a test function inspired by Gauss's estimate to solve it. This hints at something more meaningful in my Clouseau- esque accident. The setup The setup, I have come to find, is similar in spirit to Harald Cramer's probabilistic, heuristic arguments for estimating of the distribution primes (the difference here being that we don't use any other results from number theory). Here it is.. Equ. 1: The joint probability of a randomly chosen positive number n not being divisible by two relative primes p and k is (1 - 1/p)(1 - 1/k). Equ. 2: Define Q(x) = Pi(1 - 1/p) taken over all primes p <= x (Pi stands for the product symbol). Using Equ.s 1 & 2, vigorous hand waving, and a pinch of salt, we can say Equ. 3: The probability that a randomly chosen positive number x being either 1 or a prime is Q( sqrt(x) ). What we're saying here is that for x to be prime, it suffices to show that it is not divisible by any prime less than the square root of x. Equ. 3 says that in the neighborhood of x, the 'average' distance between primes is 1 / Q( sqrt(x) ). Now the n.d.e. above comes from trying to approximate Q(x) using this probabilistic model. The idea is to use that approximation in order to estimate a prime counting function: Integral] Q (sqrt(x)) dx But we don't have an analytic expression that approximates Q, yet. Instead of setting up an integral equation, we try a differential approach. Consider the change in Q as x passes over 2 very large consecutive primes p1 < p2: delta(Q) = -Q(p1) / p2 ~ -Q(x) / x delta(x) ~ 1 / Q( sqrt(x) ) Dividing the top equation by the bottom one, you get the n.d.e. I described above. The solution I had read somewhere how the 15 year old Gauss had been able to come up with his logarithmic integral for estimating the number of primes less than n. Was his integral inspired by a similar probabilistic argument? Maybe, but googling it, I couldn't find much. So, I plugged in C / ln x and solved for the constant C (=1/2). Does it mean anything? I suspect it might, which is why I posted it. I did a bit of cursory reading on the topic, but alas, I'm an amateur. My claim that Q(x) = Pi( 1 - 1/p) ~ 1 / 2 ln x does not agree with Merten's asymptotic formula (1874) Pi( 1 - 1/p) ~ exp(-C) / ln x, where C is Euler's constant. Still, there's something here that piques my nose. My result, when plugged into the prime counting integral, agrees exactly with Gauss's estimate: Integral] Q (sqrt(x)) dx ~ Integral] (1 / ln x) dx What do you think? Is this interesting, or is this old? -Babak === Subject: Re: Differential equation modeling distribution of primes posting-account=npXYCgkAAACBu5bf7OmzlG_DYSHDMhls > I have not seen a simple derivation for this estimate, and if it > exists, I am surprised why it is not more widely used in expositions > on the subject of the distribution of primes. What follows is a very > short argument based on a probability model. According to this model, > we'll find that the distribution of primes is governed by a non-linear > differential equation of the form Q'(x) = - Q(x) Q( sqrt(x) ) / x Strictly speaking, this is a _functional_ differential equation. A DE should depend only on the current value of G(x) and its derivatives. This is a _delay_ differential equation, since the current value of the derivative depend on earlier values of G. DEs tend to be associated with initial conditions that make the solution unique. On the other hand, a DDE is associated with an initial _set_ of values that makes the solution unique. whose solution is Q(x) = 1 / 2 ln x It's probably the only solution, but then, I don't know how to prove > uniqueness. (Can there be more than one solution to any given non- > linear differential equation, anyway? I suspect not.) Nonlinear differential equations typically have infinitely many solutions. An initial condition typically pins it down to a unique one. A delay differential equation will also typically have infinitely many solutions and an initial _range_ of values will typically pin it down to a unique one. Given any differentiable function q(x) on any interval of the form [a, a^2] (a > 1) that satisfies the boundary consistency condition q'(a^2) = -q(a^2)q(a)/(a^2), there is a uniquely determined solution Q of Q'(x)= -Q(x)Q(sqrt(x))/x obtained on [a^2, a^4] by integrating Q(x)'/Q(x) = -q(sqrt(x))/x . Whence, by induction, one obtains a solution for all x > a. The solution can also be extended to (1,a): obtain it on (sqrt a, a) by Q(x) = -x^2 q'(x^2)/q(x^2) and use induction again. Dan === Subject: Re: Differential equation modeling distribution of primes > Q'(x) = - Q(x) Q( sqrt(x) ) / x Strictly speaking, this is a _functional_ differential > equation. A DE should depend only on the current value of > G(x) and its derivatives. This is a _delay_ differential > equation, since the current value of the derivative depend > on earlier values of G. When x<1, sqrt(x) is a *future* time. When x=1, x=sqrt(x) ... this suggests expanding Q in powers of (x-1) ?? === Subject: Re: Differential equation modeling distribution of primes Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >we'll find that the distribution of primes is governed by a non-linear >differential equation of the form Q'(x) = - Q(x) Q( sqrt(x) ) / x See E. M. Wright, A functional equation in the heuristic theory of primes, Math. Gaz. 44 (1960), 15-16. G. Hoffman de Visme, The density of prime numbers, Math. Gaz. 45 (1961), 13-14. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: On the Complexity of Elementary Algebraic Proofs The basic idea here is that proofs which use only elementary algebra (high-school type) and induction can be considered more elementary (I use the phrase ñexceedingly elementaryî) than proofs which use arbitrary polynomials, the binomial theorem, the arithmetic-geometric mean inequality, or other concepts that involve sets of numbers where the number of items in the set can vary. This was inspired by two things: 1. my work on ñexceedingly elementaryî proofs that n^{1/n} -> 1 for large integral n; 2. the recent proof, based on the arithmetic-geometric mean inequality, that (1+1/n)^n is increasing and (1+1/n)^{n+1} is decreasing as integral n increases. The proof is ingenious: using n values of 1+1/n and 1 of n we get (1+1/(n+1))^{n+1} > (1+1/n)^n (i.e. (1+1/n)^n is increasing); using n values of 1-1/n and 1 value of 1 we get (1+1/n)^{n+1} < (1+1/(n 1))^n (i.e. (1+1/n)^{n+1} is decreasing). Since 0 < (1+1/n)^{n+1} (1+1/n)^n = (1/n)(1+1/n)^n < 4/n > 0, the limit of the two sequences exists [CapitalEth] I propose that we call it ñeî. For (1), part of it involved the following ñcontra-Bernoulliî inequality (CBI): If 0 < x < M and k is a positive integer, then (1+x)^k < 1+x*c(k,M), where c(k,M) = ((1+M)^k - 1)/M. Note: The standard BernoulliÍs inequality (BI), which is easily proved by induction, is (1+x)^n > 1+nx for x > 1 and integral n > 1 (obvious equality if and only if n = 1 or x = 0). The CBI gives an upper bound instead of a lower bound. I proved my CBI in the following way: ´ By induction, show that (1+x)^k < 1+x*c(k,M), where c(1,M)=1 and c(k +1,M) = 1+(1+M)c(k,M). ´ Again by induction, show that c(k,M) = ((1+M)^k - 1)/M. It took me a while after getting the recurrence for c(k,M) to realize what the explicit form was. This proof was done using only elementary algebra and induction, so I considered it ñexceedingly elementaryî. After another while, I realized that there was a much more direct proof: Since (1+x)^k is a polynomial with positive coefficients and constant term 1, ((1+x)^k 1)/x is also a polynomial with positive coefficients, so it is increasing, so ((1+x)^k-1)/x < ((1+M)^k-1)/M if 0 < x < M. This proof, however, seems less elementary to me, since it assumes a lot of knowledge about polynomials. Also, polynomials themselves are collections of values (the coefficients), where the number of elements in the collection can vary. For that reason, since my goal was to make as elementary a proof as possible, I preferred the first proof. I then decided to see if I could develop a more elementary proof of item (2), above. The proofs involved using the A-G mean inequality in the special case where all but one of the values were the same, so I tried to directly prove this special case. It turns out that it only needs BernoulliÍs inequality. If there are n-1 aÍs and 1 b where a and b are different and positive, we want to prove ((n-1)a+b)/n > (a^{n 1}*b)^{1/n}. We can rewrite this as a((n-1)+b/a)/n > a(b/a)^{1/n}, or, letting x = b/a, (1+(x-1)/n) > x^{1/n}. By BI, (1+(x 1)/n)^n > 1+(x-1) = x, and we are done. If we try this for n-k aÍs and k bÍs we get ((n k)a+kb)/n > (a^{n k}*b^k)^{1/n}. We can rewrite this as a((n k)+kb/a)/n > a(b/a)^{k/n}, or, letting x = b/a, (1+k(x 1)/n) > x^{k/n}. If BI were true for fractional exponents (it is, but the proof cannot be done by elementary induction), (1+k(x 1)/n)^{n/k} > 1+(x 1) = x, as before. For a proof by calculus, let n/k=a and y=x^{1/a}. The inequality becomes 1+(y^a 1)/a > y where a>1 and y>0 (equality at y=1). The values match at y=1 and the slope of the left side is y^{a-1}, which is less than 1 for y<1 and greater than 1 for y>1. This is, however, definitely non-elementary. === Subject: On different proofs of the Chevalley-Warning theorem My question is mostly a research question in the librarians' sense of the term. It is about the origins of various proofs of the Chevalley-Warning theorem. Just about the only thing I know is that Chevalley showed that if P_1(t),...,P_r(t) are polynomials in F_q[t_1,...,t_n] with zero constant term and such that sum_i deg(P_i) < n, then there exists 0 =/= x in (F_q)^n such that P_1(x) = ... = P_r(x) = 0; whereas Warning improved this by removing the assumption that they have zero constant term and concluding that the number of simultaneous solutions is divisible by p (the characteristic of F_q). Does anyone know what the original proofs of Chevalley and Warning were? In Ireland and Rosen it says that Warning's proof actually showed that the number of simultaneous solutions is at least q^{n-d} (assuming, I suppose, that there is at least one solution). Nowadays the proof that you almost invariably see is Ax's incredibly short proof, which begins with the easy observation that the number of simultaneous solutions is equal, as an element of F_q to (equivalently, is congruent mod p to) sum_{x in F_q^d} prod_{i=1}^n (1-P_i(x)^{q-1}) and then just noticing that all monomial terms appearing in this sum have the property that the sum over all x is zero. But Ireland and Rosen also give a different proof of Chevalley's part of the theorem, which I learned in my undergraduate days and has always seemed more interesting. Just now I realized that this proof can be extended to give a proof of the Warning theorem which I likewise think is more interesting than Ax's proof. I am inclined to think that my new proof is in fact not new and am asking for references. Here it goes: (I will write F for F_q to simplify notation) First a bit on reduced polynomials over F: Define a monomial c* t_1^{a_1}*...* t_n^{a_n} to be _reduced_ if a_i < q for all i. Define a polynomial to be reduced if all of its monomial terms are reduced; equivalently, it has degree strictly less than q in each of its variables. Lemma: The map which takes a (formal!) polynomial P in F[t_1,...,t_n] and returns a function Phi(P): F^n -> F_q by Phi(P)(x) = P(x) is a homomorphism of rings. Moreover: (i) The homomorphism is surjective: every function f: F^n -> F is given by some polynomial. (ii) Every polynomial is congruent modulo Ker(Phi) to a unique reduced polynomial. (iii) The kernel of Phi is the ideal . Proof: For any function f, define P_f(t) := sum_{y in F^n} f(y) prod_{i=1}^n (1-(t_i-y_i)^{q-1}). Then in the sum defining P_f(x), it is immediate to see that we get f(y)*1 when y = x and f(y)*0 otherwise. (Note that P_f is a reduced polynomial.) For (ii), evidently t_i^q - t_i is in the kernel for all i, and a finite number of applications of replacement of t_i^q in a polynomial by t_i will convert any polynomial to a reduced polynomial which induces the same function. There are q^n reduced monomials overall and these give an F-basis for the set of all reduced polynomials, so there are q^(q^n) reduced polynomials. Since this is the same as the total number of functions from F^n -> F and we know that every function is represented by a reduced monomial, the representation must therefore be unique. The deduction of (iii) from this is straightforward (and actually not used in the rest of the proof). [Remark: This lemma is fairly standard, but it seems amusing to deduce the uniqueness of the representation by reduced polynomials from the surjectivity of the evaluation map rather than the other way around.] Therefore for any function f: F^n -> F, one can speak of the reduced degree in each variable t_i and also the reduced total degree. Evidently the reduced total degree of a polynomial function P(t) is less than or equal to the degree of the polynomial (because reduction is effected by a series of operations each of which does not increase any monomial degree). Now let Z be the set of common zeros of P_1,...,P_r in F^n. Consider the characteristic function 1_Z of Z, i.e., the (F_q-valued!) function which is 1 at x if x is in Z and 0 otherwise. We have two different polynomial representations for this function. One is: P(t) = prod_{i=1}^r (1-P_i(t)^{q-1)). The other is the representation we derived above for _any_ function: in this case it is 1_Z(t) = sum_{x in F^n} prod_{i=1}^n (1-(t_i-x_i)^{q-1}). Now the total degree of P(t) is (q-1) sum_i deg(P_i) < (q-1)n, by assumption. Hence the degree of the reduced polynomial representing P is also strictly less than (q-1)n. But 1_Z is the reduced polynomial representing P, and it is a sum of polynomials each with a monomial term t_1^{q-1}*...*t_n^{q-1}. Therefore the coefficient of this monomial term in 1_Z is #Z, so unless this is zero in F_q the total degree of 1_Z is at least (q-1)*n. Done. Please let me know if you've seen such an argument before, and if so, where. === Subject: addenda There were a couple of minor typos in the preceding post: the coefficient of the final monomial is not #Z but (-1)^n #Z, and one of the last sums should be only over x in Z. I also did a bit more digging around and found some notes of Jarden on finite fields which seem to suggest that this argument might be very close to Warning's original proof! (If so, the presentation in Ireland and Rosen is then rather puzzling...) I will try to investigate this. In case anyone else has the original paper closer at hand and/or a better command of the German language, please do let me know. Pete L. Clark === Subject: Final CFP : WfPM'08, Timisoara, Romania (Sept. 29 2008) posting-account=dQDuFgoAAAAfaUMaQqHyvLwwcx_pUpxH Final CFP: Workshop on Workflow and Process Management (WfPM'08), September 2008, Timisoara, Romania EXTENDED DEADLINES ___________________________________________________________________________ [CapitalEth] ___________________ WfPM'08 -- Workshop on Workflow and Process Management in conjunction with September 29, 2008 SYNASC-2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing Timisoara, Romania September 26 - 29, 2008 Workshop Program Chairs Laura Maruster, Groningen University of Groningen, The Netherlands Teodor-Florin Fortis, West University of Timisoara, Romania Scope: The aim of WfPM'08 workshop is to bring together researchers and practitioners in workflow and business process management. This second edition of the workshop intends to identify current trends and research related with, but not restricted to, workflow modeling, workflow patterns, workflow methodologies, workflow execution engines, workflow verification and validation, interoperability, human interaction management. Topics Topics of particular interest for WfPM'08 workshop include, but are not limited to: Workflow and process modeling; Workflow methodologies; Workflow and process management; Workflows and web-services; Workflow processes and (process) reengineering technologies; Workflow verification and validation; Workflow and human interaction; Workflow integration with SOA and SOC; Workflow integration in GRID environments & GRID workflows; Workflow-based business applications; Workflow-based scientific applications; Petri nets and workflow management; Verification, validation and performance analysis of workflow management systems; Security in workflow management systems; Solutions for eSociety (eHealth, eGovernment, eLearning, etc.) The workshop, a one-day event, is a satellite event of the 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2008) that is to be held in Timisoara, Romania, last week of September 2008. Submission: Original papers reporting new results as well as experimental evaluation and comparisons of existing techniques are expected. We are requesting submissions in the form of: * Extended abstract (minimum 2 full pages) or draft research papers, describing original work; * Extended abstract or short papers (draft version), describing work in progress, system descriptions and software demonstrations. * Papers must describe original work, and should not substantially overlap with already published papers or papers that are simultaneously submitted to a journal or a conference with refereed proceedings. Full (final) research papers are limited to 8 pages (IEEE conference style found at http://synasc08.info.uvt.ro/storage/IEEE_CS_Latex.zip). Only full (final) research papers will be considered for publication in main conference post- proceedings. Papers must be submitted electronically to wfpm08[at]info.uvt.ro. Please check workshop official site at http://synasc08.info.uvt.ro/wfpm for detailed information. Workshop deadlines: Submission of papers (EXTENDED DEADLINE) (extended abstracts, draft papers): JULY 20th, 2008 Notification of acceptance (EXTENDED DEADLINE): AUGUST 1st, 2008 Full version of papers: AUGUST 25th, 2008 Final version of papers: SEPTEMBER 1st, 2008 Registration: SEPTEMBER 1st, 2008 Workshop starts: SEPTEMBER 29th, 2008 Revised papers for post-proceedings: OCTOBER 25th, 2008 Program Committee: Please check workshop official site at http://synasc08.info.uvt.ro/wfpm for detailed information. Contact: For any questions related with workshop organization, please contact organizers at one of the following addresses wfpm08[at]info.uvt.ro wfpm07[at]info.uvt.ro wfpm2007[at]gmail.ro Additional details can be found on main conference site, http://synasc08.info.uvt.ro/ === Subject: Bernoulli number identity I would like a reference for the following. I know how to prove it, for example using the generating function. But better would be to provide a reference. sum_{k=0}^n binomial(n,k) (2^k-1) B_k = 1 binomial(n,k) is a binomial coefficient n!/k!/(n-1)! Repeat: I already know how to prove it, so I am not asking for your proof. Do you know a reference for this... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Bernoulli number identity posting-account=8xHL5QoAAADvLlEAiWHpoPe_-80wlDVX I assume you mean (n-k)! rather than (n-1)!. But even so, this formula does not even work for n=0, where the left hand side yields 0. Are there other typos? On Jul 17, 1:12æpm, G. A. Edgar I know how to prove it, for example using the generating function. > But better would be to provide a reference. æ æsum_{k=0}^n binomial(n,k) (2^k-1) B_k = 1 binomial(n,k) is a binomial coefficient n!/k!/(n-1)! Repeat: I already know how to prove it, so I am not asking for your > proof. æDo you know a reference for this... -- > G. A. Edgar æ æ æ æ æ æ æ æ æ æ æ æ æ æ æhttp://www.math.ohio-state.edu/~edgar/ === Subject: Re: direct image of a locally free sheaf Originator: ilya@powdermilk.math.berkeley.edu [A complimentary Cc of this posting was sent to =?ISO-8859-1?Q?Jurgen_Bohm?= > If I am not mistaken in the following conclusions, the statement is > wrong. > I constructed a counterexample (more or less by trial and error) with > Macaulay2 by choosing A=k[a,b,c,d] and S=A[x,y], therefore X=Proj > S=P^1_A, and a certain matrix m : S^5(-1) -> S^7. So there is a sequence I tried to reduce your arguments to a little bit more (IMO) pedestrian form... a pencil is a pair of MxN matrices (A,B), or a family xA + yB; isomorphism of pencils is (A,B) |--> (QAR,QBR) with invertible Q,R; a pencil is a Jordan block if isomorphic to one with A or B equal to id, and the other one a Jordan block; a pencil is decreasing Kronecker if isomorphic to (d/dx,d/dy) acting Pol_n(x,y) --> Pol_{n-1}(x,y); increasing Kronecker is transposed; over algebraically closed fields, any pencil decomposes to a direct sum of Jordan blocks, and increasing or decreasing Kronecker blocks; the blocks appearing in the decomposition are uniquely defined; Ker and Coker(xA + yB) induce sheaves on P1 (consisting of ratios x:y); Jordan blocks correspond to skyscrapers in Coker, increasing Kronecker blocks to O(n) in Coker, decreasing to O(-n) in Ker; a family of pencils has pencils with Jordan blocks on a closed subset; a generic pencil with M=N breaks into 1x1 Jordan blocks; with M>N into M-N decreasing Kronecker blocks; n's of these blocks differ at most by one (which determines the list uniquely); a generic family of pencils has principal degeneration on: M=N: 2x2 Jordan block appears on hypersurface Disc(det(xA+yB))=0; M=N+1: 1x1 Jordan block appears on hypersurface HyperDet(A,B)=0 for N+1 x N x 2 flavor of GKZ hyperdeterminant; M>N: 1x1 Jordan block appears in codimension M-N; M=N+2, M=2m+1, m>1: a sum of two Kronecker blocks with n=m,m+1 degenerates into one with n=m-1,m+2 on a subfamily of codimension 3; M=N+2, M=2m, m>1: a sum of two Kronecker blocks with n=m degenerates into a sum with n=m-1,m+1 on a subfamily of codimension 2; [I vaguely rememeber seeing calculations of stabilizers published which would imply the last two subcases, but do not recollect where; I needed to calculate stabs explicitly... When m=1, codim decreases by 1. The M=N+1 case was observed by I.M.Gelfand/Valery???] To translate to the cases N>M, transpose. Conjecture: a generic family of rk=2 vector bundles on P1 of degree 2m+1 degenerates into O(m-1) (+) O(m+2) on codimension 3; of degree 2m degenerates into O(m-1) (+) O(m+1) on codimension 2. [These families are generically O(m) (+) O(m+1) and O(m) (+) O(m).] Conjecture: consider a family of rk=2 vector bundles on P1 parameterized by S which is isomorphic to O(n) (+) O(n') outside of subvariety S' of S, and degenerates into O(N) (+) O(N') on S'; N= 2 and n,n' >= -1, N < -1. [*] [I do not know what happens if one drops n,n' >= -1, N < -1.] [Heuristically, I would expect that if direct image is flat, so is the higher direct image (to preserve Euler characteristic). If so, then since higher direct image vanishes at generic point, it vanishes everywhere; hence the fiber of direct image is global sections of the vector bundle. But dimension of global sections jumps up on S'.] You have N=M+2, M=5, and take Coker; generically, it would be O(2) (+) O(3). You twist by O(-2); generically, the result would be O(0) (+) O(1), which in codimension 3 would degenerate into O(-1) (+) O(2) (on a closed subset it may also acquire Jordan blocks; one can ignore this). This is not in context of the conjecture above, but of [*], which for me is an unchartered territory. Following my argument, the minimal example to check would be a pencil with M=3, N=M+2: take a direct sum of Kronecker blocks with n=0,3, and take a generic 2-parameter deformation (it is enough to require that for a deformed pencil, Ker A and Ker B do not intersect). So, I would take Ker(xA + yB) with matrices A,B depending on a,b: 1 0 0 0 0 0 1 0 0 0 0 1 0 0 a 0 0 1 0 0 0 0 1 0 b 0 0 0 1 0 - and I would twist by O(1) - while to closely follow your example, the twist would be by O(2). (Since I prefer Ker to Coker, the picture is kinda dual to yours.) So O(-2) (+) O(1) would deform into O(-1) (+) O(0). Should not one be able to do it without computer algebra? :-( Hope this helps, Ilya | Determination of longitude requires precise determination of time, no > | matter how you cut it. I beg to differ. GPS depends solely on satellite position at ONE time, > NOW, and does not change from hour to hour. It is merely triangulation > no matter how you cut it. I do not need real time to calculate where > I was yesterday, I only need the positional data from the satellites > yesterday at some instant, common to all. > Time has nothing whatever to do with it, except insofar as the position > of the satellite changes as a function of time. > The ratio of the US dollar to the British pound is 4:1 (in WW II). > Time has nothing whatever to do with it, except insofar as the > ratio changes as years go by. So, to determine longitude, instead of determining the precise > position of one orbiting moon at an exact time, GPS determines > longitude by determining the exact position of several of a fleet of > orbiting satellites at an exact time. > Correct. Same problem - the difference is only a difference of degree of > difficulty. > What problem? I live within a very few miles of Flamsteed House > which has longitude zero (by definition). > http://www.jbutler.org.uk/London/Greenwich/park.shtml My present longitude is easily (!) calculated from the odometer > reading of my car, standing stationary nearby as I write. > That does NOT involve time at all. Determination of longitude DOES NOT require precise determination > of time, no matter how you cut it. Nor has it been since cables were > laid on the ocean floor to carry information, the length of which > was of greater importance than the longitude. In the past clocks were the only PRACTICAL way of determining > longitude but that is no longer true. Mariners have hung up their > astrolabes, there is no money in making them anymore. Mainly, the reason you don't need to also have a separate accurate clock to determine longitude from a GPS satellite signal... is because one of the things a GPS signal includes _is_ the time. Which is indeed compared against the expected orbital track of the satellite, so, yes, the same principle is involved. John Savard === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] | > | > | > | > | > | Determination of longitude requires precise determination of time, no | > | matter how you cut it. I beg to differ. GPS depends solely on satellite position at ONE time, | > NOW, and does not change from hour to hour. It is merely triangulation | > no matter how you cut it. I do not need real time to calculate where | > I was yesterday, I only need the positional data from the satellites | > yesterday at some instant, common to all. | > Time has nothing whatever to do with it, except insofar as the position | > of the satellite changes as a function of time. | > The ratio of the US dollar to the British pound is 4:1 (in WW II). | > Time has nothing whatever to do with it, except insofar as the | > ratio changes as years go by. So, to determine longitude, instead of determining the precise | > position of one orbiting moon at an exact time, GPS determines | > longitude by determining the exact position of several of a fleet of | > orbiting satellites at an exact time. | > Correct. Same problem - the difference is only a difference of degree of | > difficulty. | > What problem? I live within a very few miles of Flamsteed House | > which has longitude zero (by definition). | > http://www.jbutler.org.uk/London/Greenwich/park.shtml My present longitude is easily (!) calculated from the odometer | > reading of my car, standing stationary nearby as I write. | > That does NOT involve time at all. Determination of longitude DOES NOT require precise determination | > of time, no matter how you cut it. Nor has it been since cables were | > laid on the ocean floor to carry information, the length of which | > was of greater importance than the longitude. In the past clocks were the only PRACTICAL way of determining | > longitude but that is no longer true. Mariners have hung up their | > astrolabes, there is no money in making them anymore. | | Mainly, the reason you don't need to also have a separate accurate | clock to determine longitude from a GPS satellite signal... is because | one of the things a GPS signal includes _is_ the time. Which is indeed | compared against the expected orbital track of the satellite, so, yes, | the same principle is involved. The plane was over London, the ship was in the Channel, the train was in Folkstone and the car was in Dover. What time was it in New York? === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] >> | Determination of longitude requires precise determination of time, no >> | matter how you cut it. >> I beg to differ. GPS depends solely on satellite position at ONE time, >> NOW, and does not change from hour to hour. It is merely triangulation >> no matter how you cut it. I do not need real time to calculate where >> I was yesterday, I only need the positional data from the satellites >> yesterday at some instant, common to all. >> Time has nothing whatever to do with it, except insofar as the position >> of the satellite changes as a function of time. >> The ratio of the US dollar to the British pound is 4:1 (in WW II). >> Time has nothing whatever to do with it, except insofar as the >> ratio changes as years go by. So, to determine longitude, instead of determining the precise > position of one orbiting moon at an exact time, GPS determines > longitude by determining the exact position of several of a fleet of > orbiting satellites at an exact time. > Correct. Same problem - the difference is only a difference of degree of > difficulty. > What problem? I live within a very few miles of Flamsteed House > which has longitude zero (by definition). > http://www.jbutler.org.uk/London/Greenwich/park.shtml My present longitude is easily (!) calculated from the odometer > reading of my car, standing stationary nearby as I write. > That does NOT involve time at all. Until you go over a mountain and the path along the ground is longer! > Determination of longitude DOES NOT require precise determination > of time, no matter how you cut it. Nor has it been since cables were > laid on the ocean floor to carry information, the length of which > was of greater importance than the longitude. In the past clocks were the only PRACTICAL way of determining > longitude but that is no longer true. Mariners have hung up their > astrolabes, there is no money in making them anymore. > === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] | Determination of longitude requires precise determination of time, no > | matter how you cut it. I beg to differ. GPS depends solely on satellite position at ONE time, > NOW, and does not change from hour to hour. It is merely triangulation > no matter how you cut it. I do not need real time to calculate where > I was yesterday, I only need the positional data from the satellites > yesterday at some instant, common to all. > Time has nothing whatever to do with it, except insofar as the position > of the satellite changes as a function of time. > The ratio of the US dollar to the British pound is 4:1 (in WW II). > Time has nothing whatever to do with it, except insofar as the > ratio changes as years go by. So, to determine longitude, instead of determining the precise | > position of one orbiting moon at an exact time, GPS determines | > longitude by determining the exact position of several of a fleet of | > orbiting satellites at an exact time. | > Correct. Same problem - the difference is only a difference of degree of | > difficulty. | > What problem? I live within a very few miles of Flamsteed House | > which has longitude zero (by definition). | > http://www.jbutler.org.uk/London/Greenwich/park.shtml My present longitude is easily (!) calculated from the odometer | > reading of my car, standing stationary nearby as I write. | > That does NOT involve time at all. | | Until you go over a mountain and the path along the ground is longer! Greenwich Park is a mountain? In my country we call it a hill. I must remember to set my watch to mountain time so that I know where I am. Maybe Greenwich is in Colorado. | > Determination of longitude DOES NOT require precise determination | > of time, no matter how you cut it. Nor has it been since cables were | > laid on the ocean floor to carry information, the length of which | > was of greater importance than the longitude. In the past clocks were the only PRACTICAL way of determining | > longitude but that is no longer true. Mariners have hung up their | > astrolabes, there is no money in making them anymore. | > === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] > | | > | Determination of longitude requires precise determination of time, no >> | matter how you cut it. > I beg to differ. GPS depends solely on satellite position at ONE time, >> NOW, and does not change from hour to hour. It is merely triangulation >> no matter how you cut it. I do not need real time to calculate where >> I was yesterday, I only need the positional data from the satellites >> yesterday at some instant, common to all. >> Time has nothing whatever to do with it, except insofar as the position >> of the satellite changes as a function of time. >> The ratio of the US dollar to the British pound is 4:1 (in WW II). >> Time has nothing whatever to do with it, except insofar as the >> ratio changes as years go by. > | | > So, to determine longitude, instead of determining the precise > position of one orbiting moon at an exact time, GPS determines > longitude by determining the exact position of several of a fleet of > orbiting satellites at an exact time. > Correct. > | | > Same problem - the difference is only a difference of degree of > difficulty. > What problem? I live within a very few miles of Flamsteed House > which has longitude zero (by definition). > http://www.jbutler.org.uk/London/Greenwich/park.shtml > | | > My present longitude is easily (!) calculated from the odometer > reading of my car, standing stationary nearby as I write. > That does NOT involve time at all. > | > | Until you go over a mountain and the path along the ground is longer! Greenwich Park is a mountain? In my country we call it a hill. I must > remember to set my watch to mountain time so that I know where I am. > Maybe Greenwich is in Colorado. plonk === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] > | Determination of longitude requires precise determination of time, no > | matter how you cut it. I beg to differ. GPS depends solely on satellite position at ONE time, > NOW, and does not change from hour to hour. It is merely triangulation > no matter how you cut it. I do not need real time to calculate where > I was yesterday, I only need the positional data from the satellites > yesterday at some instant, common to all. > Time has nothing whatever to do with it, except insofar as the position > of the satellite changes as a function of time. > The ratio of the US dollar to the British pound is 4:1 (in WW II). > Time has nothing whatever to do with it, except insofar as the > ratio changes as years go by. So, to determine longitude, instead of determining the precise | > position of one orbiting moon at an exact time, GPS determines | > longitude by determining the exact position of several of a fleet of | > orbiting satellites at an exact time. | > Correct. Same problem - the difference is only a difference of degree of | > difficulty. | > What problem? I live within a very few miles of Flamsteed House | > which has longitude zero (by definition). | > http://www.jbutler.org.uk/London/Greenwich/park.shtml My present longitude is easily (!) calculated from the odometer | > reading of my car, standing stationary nearby as I write. | > That does NOT involve time at all. | > | | > | Until you go over a mountain and the path along the ground is longer! Greenwich Park is a mountain? In my country we call it a hill. I must | > remember to set my watch to mountain time so that I know where I am. | > Maybe Greenwich is in Colorado. | | | plonk *counter plonk* === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] >So, to determine longitude, instead of determining the precise >position of one orbiting moon at an exact time, GPS determines >longitude by determining the exact position of several of a fleet of >orbiting satellites at an exact time. Same problem - the difference is only a difference of degree of >difficulty. I disagree that it's the same problem. The practical implementation of GPS involves time, but that time is not related to the way time is used to classically determine longitude. The method used by GPS to determine location is fundamentally different, in depending on a set of artificial reference points. GPS doesn't depend on the fundamental relationship between time and longitude- it abstracts the coordinate system. (It also abstracts the time, into a serial value, not the 24-hour time.) _________________________________________________ Chris L Peterson Cloudbait Observatory http://www.cloudbait.com === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) So, to determine longitude, instead of determining the precise >position of one orbiting moon at an exact time, GPS determines >longitude by determining the exact position of several of a fleet of >orbiting satellites at an exact time. Same problem - the difference is only a difference of degree of >difficulty. I disagree that it's the same problem. The practical implementation of > GPS involves time, but that time is not related to the way time is used > to classically determine longitude. The Earth rotates beneath Foucault's pendulum at the polar axis in precisely 24 hours/360 degrees or 4 minutes for each degree of geographical seperation with the maximum value of 4 minutes equaling roughly 69 miles at the Equator. Really,really silly people like yourself can argue against the inviolate correlation between time,distance and rotation even though the principles reflected by the motion of the Earth beneath a Foucault's pendulum cannot alter.I now detest presenting the treatise of Huygens which plainly and clearly demonstrate the correlation between time,distance and planetary geometry but that remains the final authority on the matter - http://www.xs4all.nl/~adcs/Huygens/06/kort-E.html An unreasonable person can justify the rotation of the Earth beneath Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 seconds thereby cutting the correlation between 4 minutes of clock time and 1 degree of geographical seperation,that would be you Chris and the rest here.How you and your colleagues manage to do this I do not know but then again you can invent a GPS Time seperate to clock time that determines longitudes.even though you are going to have one helluva job seperating the correlation of 4 minutes for each degree of geographical seperation making 24 hours/360 degrees. It is not a matter of your muddleheaded Chris,it is the sheer volume of people who agree with you and ignore the stable reasoning of people like Huygens. The method used by GPS to determine > location is fundamentally different, in depending on a set of artificial > reference points. GPS doesn't depend on the fundamental relationship > between time and longitude- it abstracts the coordinate system. (It also > abstracts the time, into a serial value, not the 24-hour time.) > Chris L Peterson > Cloudbait Observatoryhttp://www.cloudbait.com === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] An unreasonable person can justify the rotation of the Earth beneath > Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 > seconds .... Now I understand your problem Kelleher. The earth rotates 360Á about its axis in one sidereal day. http://scienceworld.wolfram.com/astronomy/SiderealDay.html http://en.wikipedia.org/wiki/Sidereal time === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) An unreasonable person can justify the rotation of the Earth beneath > Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 > seconds .... æ æNow I understand your problem Kelleher. The earth rotates 360Á > æ æabout its axis in one sidereal day. > æ æ æhttp://scienceworld.wolfram.com/astronomy/SiderealDay.html > æ æ æhttp://en.wikipedia.org/wiki/Sidereal time After many years you understand my problem to be the demonstration of the 24 hour/360 degree correlation between clock time,planetary geometry and terrestrial longitudes,a problem I share with Huygens - And if this time of the day be the same with that observ'd where you are, then you are under the same Meridian with the place, where the Watches were set by the Sun; but if the time of the day, observ'd where you are, be greater than that shew'd by the Watches, you may be assur'd, that you are come under a more Easterly Meridian; and if less, you are come under a more Westerly. And counting for every hour of difference of time, 15 degrees of Longitude, and for every minute, 15. minutes or 1/4 of a degree, you shall then know, how many degrees, minutes, &c. the said Meridians doe differ from one another. http://www.xs4all.nl/~adcs/Huygens/06/kort-E.html You are welcome to believe that the Earth rotates through 360 degrees in 23 hours 56 minutes 04 seconds but then again you are welcome to believe in a flat Earth and insofar as it is not just the fact itself that is important but how the fact is arrived at,I will go along with Huygens,Harrison and all the rest who worked off the correlation between axial rotation,clock time,terrestrial longitudes and the daily cycle. http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png I think the Wiki representation aptly demonstrates how bad things actually are , for there you have an equable 24 hour noon cycle,equable distance from the Sun,equable orbital motion,a 3 minute 56 second differential that requires the calendar system to work,ect ect.Is it embarrassing ?.looking at all the responses is roughly the same as looking at this participant on a game show with the difference being that he does not pretend an interest in terrestrial/celestial phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html Your problem Sam is that you do not know there is a problem. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] >> Now I understand your problem Kelleher. The earth rotates 360Á >> about its axis in one sidereal day. >> http://scienceworld.wolfram.com/astronomy/SiderealDay.html >> http://en.wikipedia.org/wiki/Sidereal time After many years you understand my problem to be the demonstration of > the 24 hour/360 degree correlation between clock time,planetary > geometry and terrestrial longitudes,a problem I share with Huygens - And if this time of the day be the same with that observ'd where you > are, then you are under the same Meridian with the place, where the > Watches were set by the Sun; but if the time of the day, observ'd > where you are, be greater than that shew'd by the Watches, you may be > assur'd, that you are come under a more Easterly Meridian; and if > less, you are come under a more Westerly. And counting for every hour > of difference of time, 15 degrees of Longitude, and for every minute, > 15. minutes or 1/4 of a degree, you shall then know, how many degrees, > minutes, &c. the said Meridians doe differ from one another. http://www.xs4all.nl/~adcs/Huygens/06/kort-E.html You are welcome to believe that the Earth rotates through 360 degrees > in 23 hours 56 minutes 04 seconds but then again you are welcome to > believe in a flat Earth and insofar as it is not just the fact itself > that is important but how the fact is arrived at,I will go along with > Huygens,Harrison and all the rest who worked off the correlation > between axial rotation,clock time,terrestrial longitudes and the daily > cycle. http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png I think the Wiki representation aptly demonstrates how bad things > actually are , for there you have an equable 24 hour noon > cycle,equable distance from the Sun,equable orbital motion,a 3 minute > 56 second differential that requires the calendar system to work,ect > ect.Is it embarrassing ?.looking at all the responses is roughly the > same as looking at this participant on a game show with the difference > being that he does not pretend an interest in terrestrial/celestial > phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html Your problem Sam is that you do not know there is a problem. > What problem? Most amateur astronomers can make the measurement right from there back yards... or even out the window. Accurately sight a star against a reference anchored to the earth, such as a barn roof, utility pole, etc. Note the time of emergence. Do the same star again the next night. You say 24 hours, but I MEASURE 23:56:4.1 What do you measure Kelleher? You should try it... you might learn something new under the night sky. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) >> æ æNow I understand your problem Kelleher. The earth rotates 360Á >> æ æabout its axis in one sidereal day. >> æ æ æhttp://scienceworld.wolfram.com/astronomy/SiderealDay.html >> æ æ æhttp://en.wikipedia.org/wiki/Sidereal time After many years you understand my problem to be the demonstration of > the æ24 hour/360 degree correlation between clock time,planetary > geometry and terrestrial longitudes,a problem I share with Huygens - And if this time of the day be the same with that observ'd where you > are, then you are under the same Meridian with the place, where the > Watches were set by the Sun; but if the time of the day, observ'd > where you are, be greater than that shew'd by the Watches, you may be > assur'd, that you are come under a more Easterly Meridian; and if > less, you are come under a more Westerly. And counting for every hour > of difference of time, 15 degrees of Longitude, and for every minute, > 15. minutes or 1/4 of a degree, you shall then know, how many degrees, > minutes, &c. the said Meridians doe differ from one another. http://www.xs4all.nl/~adcs/Huygens/06/kort-E.html You are welcome to believe that the Earth rotates through 360 degrees > in 23 hours 56 minutes 04 seconds but then again you are welcome to > believe in a flat Earth and insofar as it is not just the fact itself > that is important but how the fact is arrived at,I will go along with > Huygens,Harrison and all the rest who worked off the correlation > between axial rotation,clock time,terrestrial longitudes and the daily > cycle. http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png I think the Wiki representation aptly demonstrates how bad things > actually are , for there you have an equable 24 hour noon > cycle,equable distance from the Sun,equable orbital motion,a 3 minute > 56 second differential that requires the calendar system to work,ect > ect.Is it embarrassing ?.looking at all the responses is roughly the > same as looking at this participant on a game show with the difference > being that he does not pretend an interest in terrestrial/celestial > phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html Your problem Sam is that you do not know there is a problem. æ æWhat problem? > The crisis is not knowing there is a problem whereas the resolution of the problem is dealing with the matter.You belong to the former and cannot be helped and are welcome to believe in things that reasonable people generally would not however it still leaves me to explain why Flamsteed was wrong in jumping to a stupid conclusion One of the major problems is that it is not even difficult to disprove. For a star to return to a location 3 mninutes 56 seconds earlier without fail requires the calendar saystem to work,the convenient system which collects the fraction of days based on 365 days 5 hours 49 minutes and reworks it into a system of 3 years of 365 daysd and 1 year of 366 days. Now somewhere at the bottom of your brain you probably have some sort of inkling that the calendar system is not a good way to describe the annual orbital motion of the Earth .but then again,you probably have convinced yourself that the 3 minutes 56 second difference does not need the 365/366 day system.What person here,at least who have the ability to think for themselves,can find a way to justify the 3 minute 56 second difference without the need of the leap day every 4th year ?.The answer is that you cannot and that is why the representation,the reasoning and the phony value is a fraud - http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png > æ æMost amateur astronomers can make the measurement right from there > æ æback yards... or even out the window. Accurately sight a star against > æ æa reference anchored to the earth, such as a barn roof, utility pole, > æ æetc. Note the time of emergence. Do the same star again the next night. > And the next night and the next night andf when it comes to Feb 29th in the 4th year it will still return 3 minutes 56 seconds earlier just like the night before because the reasoning is calendrically based !!.Do you really believe you are basing the orbital motion of the Earth on 365 days 5 hours 49 minutes when you are actually basing it on the calendar system of equable days .If you did not insert the leap day correction,you 3 minute 56 second correlation would fail and that is why the whole scheme is fraudulent. > æ æYou say 24 hours, but I MEASURE 23:56:4.1 æ æWhat do you measure Kelleher? æYou should try it... you might learn > æ æsomething new under the night sky.- Hide quoted text - - Show quoted text - Using a starreturning 3 minutes 56 seconds earlier as a means for justifying axial and orbital motion of the Earth is no better or worse than a flat Earth.The crisis exists in not knowing there is a problem but after that it becomes a matter of dealing with it.If you cannot raise your reasoning to the point of view of the 3 minutes 56 second difference expressed against the calendar system then you are perfectly entitled to believe whatever you want. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=1nOeKQkAAABD2jxp4Pzmx9Hx5g9miO8y Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > What problem? Most amateur astronomers can make the measurement right from there > back yards... or even out the window. Accurately sight a star against > a reference anchored to the earth, such as a barn roof, utility pole, > etc. Note the time of emergence. Do the same star again the next night. You say 24 hours, but I MEASURE 23:56:4.1 What do you measure Kelleher? You should try it... you might learn > something new under the night sky. Ah, but in his posts, he has explained that using the stars as a standard is wrong-headed; it is astrological geometry! He believes the return of a star - which he is happy to admit happens in 23 hours, 56 minutes, and 4 seconds - is _irrelevant_ to the true axial rotation of the Earth, which can only be determined by a completely different approach. But even he doesn't know what that approach is, since astronomy isn't that advanced yet (it stagnated after Kepler because Newton confused everybody). John Savard === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] > What problem? >> Most amateur astronomers can make the measurement right from there >> back yards... or even out the window. Accurately sight a star against >> a reference anchored to the earth, such as a barn roof, utility pole, >> etc. Note the time of emergence. Do the same star again the next night. >> You say 24 hours, but I MEASURE 23:56:4.1 >> What do you measure Kelleher? You should try it... you might learn >> something new under the night sky. Ah, but in his posts, he has explained that using the stars as a > standard is wrong-headed; it is astrological geometry! He believes the > return of a star - which he is happy to admit happens in 23 hours, 56 > minutes, and 4 seconds - is _irrelevant_ to the true axial rotation of > the Earth, which can only be determined by a completely different > approach. But even he doesn't know what that approach is, since astronomy isn't > that advanced yet (it stagnated after Kepler because Newton confused > everybody). John Savard Aaaaah, so Kelleher is really a true looony! === Subject: Re: Could the Antikythera have been used to determine longitude?[was > æ æWhat problem? > æ æMost amateur astronomers can make the measurement right from there >> æ æback yards... or even out the window. Accurately sight a star against >> æ æa reference anchored to the earth, such as a barn roof, utility pole, >> æ æetc. Note the time of emergence. Do the same star again the next night. > æ æYou say 24 hours, but I MEASURE 23:56:4.1 > æ æWhat do you measure Kelleher? æYou should try it... you might learn >> æ æsomething new under the night sky. Ah, but in his posts, he has explained that using the stars as a > standard is wrong-headed; it is astrological geometry! He believes the > return of a star - which he is happy to admit happens in 23 hours, 56 > minutes, and 4 seconds - is irrelevant to the true axial rotation of > the Earth, which can only be determined by a completely different > approach. But even he doesn't know what that approach is, since astronomy isn't > that advanced yet (it stagnated after Kepler because Newton confused > everybody). John Savard æ æAaaaah, so Kelleher is really a true looony!- Hide quoted text - - Show quoted text - No problem Sam,you are perfectly entitled to believe what you want and if you see a star returning 3 minutes 56 seconds earlier tonight than last night and believe that this represents axial rotation and orbital motion then good for you - http://upload.wikimedia.org/wikipedia/commons/1/1d/Tiempo sid%C3%A9reo.en.png Maybe you can stick a smiley face on the Sun to complete the cartoon conception for the Earth's axial and orbital motion but as a matter of astronomical substance,you may as well believe in a flate Earth. Looking at the matter from the point of view of a 3 minute 56 second difference that never fails and seeing that it needs the equable 365/366 days of the calendar system to work,we leave the realm of astronomy and enter the cartoon world of astrologers .I enjoyed the challenge of working out why Flamsteed was wrong by dirtectly linking axial rotation and subsequently orbital motion to the return of a star but again,that is no longer astronomy. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=LrLGLAkAAAAGjouKUEM9rprSVVG_OeJd Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) .looking at all the responses is roughly the > same as looking at this participant on a game show with the difference > being that he does not pretend an interest in terrestrial/celestial > phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html > Wow. That poor game show host. It was like he was aching to jump in and say It's the Moon. The Moon, that rotates around the Earth! What's even worse than the contestant getting the question wrong was that 56% of the audience members said the Sun and only 42% said the Moon. I guess we shouldn't feel so bad about the poor grade school education in the U.S. Bob Clark === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) > .looking at all the responses is roughly the same as looking at this participant on a game show with the difference > being that he does not pretend an interest in terrestrial/celestial > phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html æWow. That poor game show host. It was like he was aching to jump in > and say It's the Moon. The Moon, that rotates around the Earth! > æWhat's even worse than the contestant getting the question wrong was > that 56% of the audience members said the Sun and only 42% said the > Moon. I guess we shouldn't feel so bad about the poor grade school > education in the U.S. æ æBob Clark 100% of participants here who profess an interest in astronomy do not know how clocks are kept in sync with the axial; cycle at 24 hours 360 degrees without the need of an external cyclical reference or believe that the Earth rotates to noon every 24 hours - http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png Whatever you may think of the contestant on the game show,you are far worse by virtue of the value for axial rotation through 360 degrees and the reasoning which leads you to that value. Of course you have no sense that there is a problem and that is where the crisis is. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] | .looking at all the responses is roughly the | > same as looking at this participant on a game show with the difference | > being that he does not pretend an interest in terrestrial/celestial | > phenomena - http://www.maniacworld.com/pitiful-answer-on-game-show.html | > | | Wow. That poor game show host. It was like he was aching to jump in | and say It's the Moon. The Moon, that rotates around the Earth! | What's even worse than the contestant getting the question wrong was | that 56% of the audience members said the Sun and only 42% said the | Moon. I guess we shouldn't feel so bad about the poor grade school | education in the U.S. | | Bob Clark | Now you see what the metric system is REALLY for, those frogs can only count if they have enough fingers. And to think King Louis Pasteur XIV taught that stupid Italian, Galileo, and that idiot German, Kepler, all about the geocentric system.... Or was it Copernicus... === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) | .looking at all the responses is roughly the > same as looking at this participant on a game show with the difference > being that he does not pretend an interest in terrestrial/celestial > phenomena - > | | >http://www.maniacworld.com/pitiful-answer-on-game-show.html > | | > | Wow. That poor game show host. It was like he was aching to jump in > | and say It's the Moon. The Moon, that rotates around the Earth! > | What's even worse than the contestant getting the question wrong was > | that 56% of the audience members said the Sun and only 42% said the > | Moon. I guess we shouldn't feel so bad about the poor grade school > | education in the U.S. > | > | æ Bob Clark > | > Now you see what the metric system is REALLY for, those frogs > can only count if they have enough fingers. æAnd to think King Louis > Pasteur XIV taught that stupid Italian, Galileo, and that idiot German, > Kepler, all about the geocentric system.... Or was it Copernicus... Galileo knew all too well that people can adopt the worse possible views but it does not happen that such people gain dominance like the strangehold structural astrologers have now on astronomy.How it comes to be that heliocentric reasoning remains in ruins while astrological conceptions flourish hardly compares with the sheer gloating that no objections are raised to the most ridiculous of notions suchg as the 'sidereal time' justification for axial and orbital motion. Maybe an intelligent person can acknowledge what Galileo is saying and then identify themselves meaninglfully by fighting these false ideologies which are so intellectually poor as to be almost sub- human . Here you go John,remember,you are among those whom Galileo finds objectionable because when you put your sticks in the ground and justify axial rotation in 23 hours 56 minutes 04 seconds,no amount of reasoning can convince you or the rest here how utterly silly that it - SALV. The same thing has struck me even more forcibly than you. I have heard such things put forth as I should blush to repeat--not so much to avoid discrediting their authors (whose names could always be withheld) as to refrain from detracting so greatly from the honor of the human race. In the long run my observations have convinced me that some men, reasoning preposterously, first establish some conclusion In their minds which, either because of its being their own or because of their having received it from some person who has their entire confidence, impresses them so deeply that one finds it impossible ever to get it out of their heads. Such arguments in support of their fixed idea as they hit upon themselves or hear set forth by others, no matter how simple and stupid these may be, gain their instant acceptance and applause. On the other hand whatever is brought forward against it, however ingenious and conclusive, they receive with disdain or with hot rage--if indeed it does not make them ill. Beside themselves with passion, some of them would not be backward even about scheming to suppress and silence their adversaries. I have had some experience of this myself. SAGR. I know; such men do not deduce their conclusion from its premises or establish it by reason, but they accommodate (I should have said discommode and distort) the premises and reasons to a conclusion which for them is already established and nailed down. No good can come of dealing with such people, especially to the extent that their company may be not only unpleasant but dangerous. Therefore let us continue with our good Simplicio, who has long been known to me as a man of great ingenuity and entirely without malice. Besides, he is intimately familiar with the Peripatetic doctrine, and I am sure that whatever he does not think up in support of Aristotle's opinion is not I likely to occur to anybody. Dialogue Concerning the Two Chief World Systems, 1632 Galileo Start with an idea that the Earth axial rotation is 23 hours 56 minutes 04 seconds and everything else becomes impossible. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=vyjRTQgAAAAlmipmdUnA2I7cIRBWlv7_ Gecko/20061029 SeaMonkey/1.0.6,gzip(gfe),gzip(gfe) ---snip--- Start with an idea that the Earth axial rotation is 23 hours 56 > minutes 04 seconds and everything else becomes impossible. OK I create a new time standard. On this date, I create a clock that has 64 Caperns per axial revolution. 64 is a nice power of two. Now, how do I know when to go to work? How do I know when to bring out my telescope to observe planet or stars? How do I know when to plant my crops? I reference this to the most accurate clock I can find but as the years go by, the rotation of the Earth slows slightly and it is no longer in synch with the clocks. What do I do now? What did I do wrong? Why is my clock no better than the original 24 solar hour clock? Save me Gerald!! Dwight === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=ga13VwoAAACkP2GXOrdrjILWKVDk7adJ CLR 1.1.4322),gzip(gfe),gzip(gfe) > An unreasonable person can justify the rotation of the Earth beneath > Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 > seconds .... æ æNow I understand your problem Kelleher. The earth rotates 360Á > æ æabout its axis in one sidereal day. > æ æ æhttp://scienceworld.wolfram.com/astronomy/SiderealDay.html > æ æ æhttp://en.wikipedia.org/wiki/Sidereal time > You are welcome to believe that the Earth rotates through 360 degrees > in 23 hours 56 minutes 04 seconds Your problem Sam is that you do not know there is a problem. Take the Sun out of the picture all together, and set the meridian reference for a 24 hour rotational period to a (much more) distant of rotation were insufficient to maintain the solar day. You would have to add time to keep the Sun on the meridian at the same time each day. That time would be 3 minutes 56 seconds. Then every four years, you'd have to subtract a day to fix the calendar. Worse, you'd have to try to calculate wall clock time based on a 24 hour 3 minute 56 second solar day. I like the 24 hour clock day for simplicity sake, and the 23 hour 56 minutes 4 seconds sidereal day, so that I can figure out what stars will be visible in the night sky tonight. Rotation schmotation. Just pick a standard and stick with it. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) An unreasonable person can justify the rotation of the Earth beneath > Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 > seconds .... æ æNow I understand your problem Kelleher. The earth rotates 360Á > æ æabout its axis in one sidereal day. > æ æ æhttp://scienceworld.wolfram.com/astronomy/SiderealDay.html > æ æ æhttp://en.wikipedia.org/wiki/Sidereal time You are welcome to believe that the Earth rotates through 360 degrees > in 23 hours 56 minutes 04 seconds Your problem Sam is that you do not know there is a problem. Take the Sun out of the picture all together, and set the meridian > reference for a 24 hour rotational period to a (much more) distant > of rotation were insufficient to maintain the solar day. > Let me give you lesson in how a true astronomer approaches this and how it actually happens that the correlation between clocks,terrestrial longitudes and the rotational cycle remain inviolate despite the utter stupidity that attaches itself to tyhe value of 23 hours 56 minutes 04 seconds. The Equation of Time represents a very,very old astronomical principle that the noopn cycles are observed to be unequl despite the dismal belief that the noon cycoles are equal among those who chase the 'sidereal time' rainbow - http://en.wikipedia.org/wiki/Image:Tiempo sid%C3%A9reo.en.png The Equation opf Time equalises the variations in the natural noon cycle to an equyable 24 hour cycle as a weighed average against the annual orbital cycle.This is difinitiove and if you want to hear it from Huygens then here it is - Here take notice, that the Sun or the Earth passeth the 12. Signes, or makes an entire revolution in the Ecliptick in 365 days, 5 hours 49 min. or there about, and that those days, reckon'd from noon to noon, are of different lenghts; as is known to all that are vers'd in Astronomy http://www.xs4all.nl/~adcs/Huygens/06/kort-E.html There are no loopholes,there are no escape clauses,the natural noon cycle is unequal,the 24 hour cycle is a human devised primciple which not only creates the 24 hour cycle but keeps these cycles elapsing seamlessly into each other.Because this explanationi definitive,I will write in capital letters the two most important points as an exception in all my years posting on the usenet - THERE IS NO EXTERNAL CYCLICAL REFERENCE FOR THE AVERAGE 24 HOUR CYCLE . When Copernicus resolved the apparent retrograde motion of the other planets by assigning an orbital motion to the Earth it left axial rotation to explain the daily cycle.The brilliant,do you hear,the brilliant maneuver of the astronomical timekeepers was to transfer the Equation of Time principles which create the 'average' 24 hour day to axial rotation as a 'constant'.,hence important poiunt nuimber two which meshes with the other important point - THERE IS NO EXTERNAL CYCLICAL REFERENCE FOR CONSTANT AXIAL ROTATION The average 24 hour day is broken into hours, minutes, and seconds which can be divided longitudionally through 360 degrees where 4 minutes equal 1 degree of longitude and 24 hours /360 degrees in total.The Equation of Time principles maintain the 24 hour day,keepo the cycles constant,transfer it to the average/constant axial cycle as a convenience but not as an observed 'fact' as the siderealists or celestial sphere astrologers try to do. > You would have to add time to keep the Sun on the meridian at the same > time each day. That time would be 3 minutes 56 seconds. Then every > four years, you'd have to subtract a day to fix the calendar. Worse, > you'd have to try to calculate wall clock time based on a 24 hour 3 > minute 56 second solar day. > In the end you wqill still believe that a location rotates through 360 degrees in 23 hours 56 minutes 04 seconds and I woild have nothing to gain by being seen to argue at such a low level. > I like the 24 hour clock day for simplicity sake, and the 23 hour 56 > minutes 4 seconds sidereal day, so that I can figure out what stars > will be visible in the night sky tonight. Rotation schmotation. Just pick a standard and stick with it.- Hide quoted text - - Show quoted text - Good for you. === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] posting-account=I3BSGAoAAADBa0hVQRltLB3YrdNlBH-l InfoPath.1),gzip(gfe),gzip(gfe) An unreasonable person can justify the rotation of the Earth beneath > Foucault's pendulum at the polar axis in 23 hours 56 minutes 04 > seconds .... æ æNow I understand your problem Kelleher. The earth rotates 360Á > æ æabout its axis in one sidereal day. > æ æ æhttp://scienceworld.wolfram.com/astronomy/SiderealDay.html > æ æ æhttp://en.wikipedia.org/wiki/Sidereal time The Wiki representation is incredible in brealing a dozen different astronomical principles such as an equable 24 hour noon cycle,equable distance from the Sun,equable orbital motion,a 3 minute 56 second differential that req === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] > So, to determine longitude, instead of determining the precise >> position of one orbiting moon at an exact time, GPS determines >> longitude by determining the exact position of several of a fleet of >> orbiting satellites at an exact time. >> Same problem - the difference is only a difference of degree of >> difficulty. I disagree that it's the same problem. The practical implementation of > GPS involves time, but that time is not related to the way time is used > to classically determine longitude. The method used by GPS to determine > location is fundamentally different, in depending on a set of artificial > reference points. GPS doesn't depend on the fundamental relationship > between time and longitude- it abstracts the coordinate system. (It also > abstracts the time, into a serial value, not the 24-hour time.) > _________________________________________________ Chris L Peterson > Cloudbait Observatory > http://www.cloudbait.com Just For Reference: http://www.colorado.edu/geography/gcraft/notes/gps/gif/navigate.gif http://www.colorado.edu/geography/gcraft/notes/gps/gif/gdop.gif http://www.colorado.edu/geography/gcraft/notes/gps/gps.html === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] >I beg to differ. GPS depends solely on satellite position at ONE time, >NOW, and does not change from hour to hour. It is merely triangulation >no matter how you cut it. I do not need real time to calculate where >I was yesterday, I only need the positional data from the satellites >yesterday at some instant, common to all. >Time has nothing whatever to do with it, except insofar as the position >of the satellite changes as a function of time. I assume you're being deliberately obtuse. Obviously there are ways of determining longitude without worrying about absolute time. Building an external set of references, as with the GPS system, is one of those. But before we could do that, the only way to determine longitude was to compare local time to the time a fixed reference location. For virtually all of recorded history, determining longitude was equivalent to knowing the time. _________________________________________________ Chris L Peterson Cloudbait Observatory http://www.cloudbait.com === Subject: Re: Could the Antikythera have been used to determine longitude?[was Re: SWEDENBORG ON LONGITUDE] | | >I beg to differ. GPS depends solely on satellite position at ONE time, | >NOW, and does not change from hour to hour. It is merely triangulation | >no matter how you cut it. I do not need real time to calculate where | >I was yesterday, I only need the positional data from the satellites | >yesterday at some instant, common to all. | >Time has nothing whatever to do with it, except insofar as the position | >of the satellite changes as a function of time. | | I assume you're being deliberately obtuse. Well done, you managed to assume. | Obviously there are ways of | determining longitude without worrying about absolute time. That's what I said and gave an example, obviously. I assume you are a cretin attempting to be obtuse but not succeeding. === Subject: Group generator If p is a prime then ({1,...,p-1},*_p) is a cyclic group. What is the generator of that group? What is the generator of Eulers group for any natural number n? === Subject: Re: Group generator > If p is a prime then ({1,...,p-1},*_p) is a cyclic group. > What is the generator of that group? Look up primitive root. > What is the generator of Eulers group for any natural number n? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Group generator posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > If p is a prime then ({1,...,p-1},* p) is a cyclic group. > What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? Better yet, try to find all the elements which are generators for, say, p <= 17. Maybe when you see what they are you will be able to make an educated guess? > What is the generator of Eulers group for any natural number n? What's Euler's group? Are you sure it is cyclic for all n? Have you tried finding generators for it for small n? -- m === Subject: Re: Group generator >> If p is a prime then ({1,...,p-1},*_p) is a cyclic group. >> What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? Well I did find some generators for some small p and it is usually (p+1)/2. I tried to assume that when p!=2^n-1 the generator is (p+1)/2 however I was not able to prove it. If anyone knows the answer please tell me. === Subject: Re: Group generator >> If p is a prime then ({1,...,p-1},*_p) is a cyclic group. >> What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? Well I did find some generators for some small p and it is usually (p+1)/2. > I tried to assume that when p!=2^n-1 the generator is (p+1)/2 however I was > not able to prove it. If anyone knows the answer please tell me. The answer is that there is no answer. There is no formula that gives you a generator of the multiplicative group modulo p. All you can do in practice is check to see whether 2 is a generator, and, if it isn't, try 3, then 5, 6, etc., until you find one that works. Well, there may be some shortcuts, but in any event there's no such thing as this simple formula always works. Incidentally, this is a good way to demonstrate the distinction between isomorphism and equality. The additive group mod 100 and the multiplicative group mod 101 are isomorphic, as each is cyclic of order 100. But they aren't the same, as it's trivial to find a generator in one case, and not in the other. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Group generator days. My association with the Department is that of an alumnus. > If p is a prime then ({1,...,p-1},*_p) is a cyclic group. > What is the generator of that group? >> `The' generator? Usually, a group has many generators... >> Have you tried finding a generator by hand in the case of small p? >> Better yet, try to find all the elements which are generators >> for, say, p <= 17. Maybe when you see what they are you will >> be able to make an educated guess? Well I did find some generators for some small p and it is usually (p+1)/2. >I tried to assume that when p!=2^n-1 the generator is (p+1)/2 however I was >not able to prove it. If anyone knows the answer please tell me. p 248, problem F9: A primitive root g of a prime p is a number such that the residudue classes of g, g^2, ..., g^{p-1}=1 are all distinct. For example, 5 is a primitive root modulo 23[.] There is a famous conjecture of Artin that for each integer g =/= -1, g not a square, there are infinitely many primes p with g as a primitive root. Hooley proved this assuming the extended Riemann hypothesis, Gupta & Murty proved it unconditionally for infinitely many g. Heath-Brown proved that for at most two exceptional primes p_1, p_2, the following is true: for each prime p, there are infinitely many primes q with p as a primitive root of q. -- 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: Group generator On Jul 21, 10:18æam, Mariano Su.87rez-Alvarez If p is a prime then ({1,...,p-1},* p) is a cyclic group. > What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? What is the generator of Eulers group for any natural number n? What's Euler's group? Are you sure it is cyclic for all n? > Have you tried finding generators for it for small n? -- m I could have also pointed to: ~A === Subject: Re: Group generator posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 21, 10:18æam, Mariano Su.87rez-Alvarez If p is a prime then ({1,...,p-1},* p) is a cyclic group. > What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? What is the generator of Eulers group for any natural number n? What's Euler's group? Are you sure it is cyclic for all n? > Have you tried finding generators for it for small n? -- m Mariano - very good response! I'll add a bit more by pointing the OP to - Good luck! ~A === Subject: Re: Group generator posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > On Jul 21, 10:18æam, Mariano Su.87rez-Alvarez > If p is a prime then ({1,...,p-1},* p) is a cyclic group. > What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? What is the generator of Eulers group for any natural number n? What's Euler's group? Are you sure it is cyclic for all n? > Have you tried finding generators for it for small n? -- m Mariano - very good response! It makes my heart ache when I see people acting as if mathematical truth is communicated to the select few by messengers from the gods or something... When one opens Gauss's disquisitiones, it is plain to the eye to see that a good 2/3s of it are essentially notes on the procedures he used to actually compute examples; and that was Gauss himself! It is a very weird turn of events that lots of people studying such matters now do not even think of computing an example. We have computers now! The tables of class numbers, for example, that took Kummer a *life* to compute we can compute in seconds. Yet taking p = 11 and finding by hand the generating elements of Z p is beyond most of out students... It is a relief to find out that quite a few students actually get on with the program when they notice that you answer 3/4th of their questions by `can you give me an example'? -- m === Subject: Re: Group generator >It makes my heart ache when I see people acting as >if mathematical truth is communicated to the select >few by messengers from the gods or something... When one opens Gauss's disquisitiones, it is >plain to the eye to see that a good 2/3s of it are >essentially notes on the procedures he used to actually >compute examples; and that was Gauss himself! The law of quadratic reciprocity is perhaps a good example of something that looks like it was handed down from the gods, but which fairly leaps out at you off the page if you slog through some computations of quadratic residues for small prime moduli. The god-like form is (p|q)(q|p) = (-1)^{(p-1)(q-1)/4} - what mortal would ever think of that? - but the form you are likely to guess yourself after doing a bit of grunt work is that, if a is a positive integer, and p and q are two odd primes such that p = (+/-)q (mod 4a), then (a|p) = (a|q). References (chapters 4 and 7 respectively): @book{adams-gold:number, author = William W. Adams and Larry Joel Goldstein, title = Introduction to Number Theory, publisher = Prentice-Hall, Inc., year = 1976} @book{ash-gross:symmetry, author = Avner Ash and Robert Gross, title = Fearless Symmetry: Exposing the Hidden Patterns of Numbers, publisher = Princeton University Press, year = 2006} (I haven't got the Adams and Goldstein book to hand, but I have a note that it gives the derivation, which is straightforward - unlike the full proof of the law itself, of course.) -- === Subject: Re: Group generator <4q2a84ps0h6s7ab3suek93csdvdnr8ih2r@4ax.com> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) It makes my heart ache when I see people acting as >if mathematical truth is communicated to the select >few by messengers from the gods or something... When one opens Gauss's disquisitiones, it is >plain to the eye to see that a good 2/3s of it are >essentially notes on the procedures he used to actually >compute examples; and that was Gauss himself! The law of quadratic reciprocity is perhaps a good example of > something that looks like it was handed down from the gods, but > which fairly leaps out at you off the page if you slog through > some computations of quadratic residues for small prime moduli. Indeed. In fact, I contend that *very* few results are the result of a god descending to earth and communicating a statement ;-) -- m === Subject: Re: Group generator >Indeed. In fact, I contend that *very* few >results are the result of a god descending >to earth and communicating a statement ;-) Ramanujan being the main exception? -- === Subject: Re: Group generator posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 21, 10:52æam, Mariano Su.87rez-Alvarez On Jul 21, 10:18æam, Mariano Su.87rez-Alvarez > If p is a prime then ({1,...,p-1},* p) is a cyclic group. > What is the generator of that group? `The' generator? Usually, a group has many generators... Have you tried finding a generator by hand in the case of small p? > Better yet, try to find all the elements which are generators > for, say, p <= 17. Maybe when you see what they are you will > be able to make an educated guess? What is the generator of Eulers group for any natural number n? What's Euler's group? Are you sure it is cyclic for all n? > Have you tried finding generators for it for small n? -- m Mariano - very good response! It makes my heart ache when I see people acting as > if mathematical truth is communicated to the select > few by messengers from the gods or something... When one opens Gauss's disquisitiones, it is > plain to the eye to see that a good 2/3s of it are > essentially notes on the procedures he used to actually > compute examples; and that was Gauss himself! It is a > very weird turn of events that lots of people studying > such matters now do not even think of computing an example. > We have computers now! The tables of class numbers, for > example, that took Kummer a *life* to compute we can compute > in seconds. Yet taking p = 11 and finding by hand > the generating elements of Z p is beyond most > of out students... It is a relief to find out that > quite a few students actually get on with the program > when they notice that you answer 3/4th of their questions > by `can you give me an example'? -- m I totally agree with your comments. I used to use Mathematica to investigate all sorts of items - particularly in Number Theory. The students today have all these wonderful tools at the disposal - and it makes learning so much more interesting, fruitful and experimental. I wish the schools would spend much more time convincing students to learn to experiment, question, validate and explore with all of the wonderful CAS programs available on the market (and many free ones). This gets them prototyping, programming and exploring mathematics - so they can see the beauty and also to expand the problem solving abilities. ~A === Subject: Re: An exact simplification challenge - 66 (MeijerG) posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) MeijerG[{{-1, 0, 1/2}, {}}, {{-1/4, 0}, {-3/4}}, 1] > You can try also: MeijerG([[], []],[[89/55, 78/55, 67/55, 56/55, 9/11], []],10/7) MeijerG([[], []],[[113/60, 101/60, 89/60, 77/60, 13/12], []],1/2) MeijerG([[], []],[[93/85, 76/85, 59/85, 42/85, 5/17], []],2) MeijerG([[], []],[[73/60, 61/60, 49/60, 37/60, 5/12], []],4) MeijerG([[], []],[[53/60, 41/60, 29/60, 17/60, 1/12], []],9/14) MeijerG([[], []],[[5/3, 22/15, 19/15, 16/15, 13/15], []],17/9) MeijerG([[], []],[[138/85, 121/85, 104/85, 87/85, 14/17], []],4/3) MeijerG([[], []],[[93/85, 76/85, 59/85, 42/85, 5/17], []],2) MeijerG([[], []],[[37/15, 34/15, 31/15, 28/15, 5/3], []],1/4) MeijerG([[], []],[[71/45, 62/45, 53/45, 44/45, 7/9], []],1/9) MeijerG([[], []],[[88/85, 71/85, 54/85, 37/85, 4/17], []],9/5) MeijerG([[], []],[[29/5, 28/5, 27/5, 26/5, 5], []],13/2) MeijerG([[], []],[[24/5, 23/5, 22/5, 21/5, 4], []],5) MeijerG([[], []],[[23/10, 21/10, 19/10, 17/10, 3/2], []],1/6) MeijerG([[], []],[[111/20, 107/20, 103/20, 99/20, 19/4], []],1/2) MeijerG([[], []],[[104/55, 93/55, 82/55, 71/55, 12/11], []],7/17) MeijerG([[], []],[[73/35, 66/35, 59/35, 52/35, 9/7], []],10/9) MeijerG([[], []],[[166/95, 147/95, 128/95, 109/95, 18/19], []],6) MeijerG([[], []],[[78/35, 71/35, 64/35, 57/35, 10/7], []],1/2) MeijerG([[], []],[[72/65, 59/65, 46/65, 33/65, 4/13], []],13/5) MeijerG([[], []],[[1, 4/5, 3/5, 2/5, 1/5], []],3/7) MeijerG([[], []],[[67/40, 59/40, 51/40, 43/40, 7/8], []],3/8) MeijerG([[], []],[[116/45, 107/45, 98/45, 89/45, 16/9], []],9/14) MeijerG([[], []],[[82/15, 79/15, 76/15, 73/15, 14/3], []],17/15) MeijerG([[], []],[[136/95, 117/95, 98/95, 79/95, 12/19], []],3/8) MeijerG([[], []],[[88/85, 71/85, 54/85, 37/85, 4/17], []],17/6) MeijerG([[], []],[[41/20, 37/20, 33/20, 29/20, 5/4], []],17/4) MeijerG([[], []],[[22/15, 19/15, 16/15, 13/15, 2/3], []],8/7) MeijerG([[], []],[[23/10, 21/10, 19/10, 17/10, 3/2], []],10) MeijerG([[], []],[[137/65, 124/65, 111/65, 98/65, 17/13], []],9/5) MeijerG([[], []],[[141/70, 127/70, 113/70, 99/70, 17/14], []],4/5) MeijerG([[], []],[[11/5, 2, 9/5, 8/5, 7/5], []],19/3) MeijerG([[], []],[[9/5, 8/5, 7/5, 6/5, 1], []],20/9) MeijerG([[], []],[[31/20, 27/20, 23/20, 19/20, 3/4], []],1/6) MeijerG([[], []],[[54/5, 53/5, 52/5, 51/5, 10], []],14/3) MeijerG([[], []],[[17/15, 14/15, 11/15, 8/15, 1/3], []],4) MeijerG([[], []],[[13/5, 12/5, 11/5, 2, 9/5], []],1/3) MeijerG([[], []],[[73/35, 66/35, 59/35, 52/35, 9/7], []],13/2) MeijerG([[], []],[[17/15, 14/15, 11/15, 8/15, 1/3], []],1/2) MeijerG([[], []],[[23/10, 21/10, 19/10, 17/10, 3/2], []],2/3) === Subject: On Penrose's argument against density operators posting-account=EI6PUAoAAAAazrTcDkROhitOfD8_tWFT 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) http://cmathphil.blogspot.com/2008/07/on-penroses-argument-against-density.h tml the usage of the density operator when dealing with pure quantum states. The main arguments of Penrose are reviewed, and the reflection proceeds around the main question: What is the most fundamental mathematical structure that should be used to describe the quantum system, and, what is the nature of the physical semantics that this structure formalizes? formulas that appear as images (that can be seen in the above link): On Penrose's Argument Against Density Operators by Carlos Pedro Gon.8dalves What is a quantum state? Should we speak of a quantum state at all? Should we speak of quantum states or of quantum processes? These questions can be raised from the work of Baugh, Finkelstein and Galiautdinov (http://arxiv.org/abs/hep-th/0206036) and from the results obtained by Gon.8dalves and Madeira (http://www.ma.utexas.edu/ mp arc/c/08/08-62.pdf), about the connection between a stationary quantum state and the consistent histories formalism, these results being obtained from the relational structure of the different bases in which a stationary quantum state can be expanded. A different, but related, problem arises from PenroseÍs Road to structure that should be used to formalize what is usually called a quantum state. Thinking about these two threads, one is lead to the following question: What is the most fundamental mathematical structure that should be used to describe the quantum system, and, what is the nature of the physical semantics that this structure formalizes? This is the main question to which we shall return, recurrently, formalizing the quantum state by: A) The density operator, whose entropy zero space (using von NeumannÍs notion of entropy) is comprised of the density operators for the so- called pure states. B) The normalized kets (for the pure states) It is clear that the density operator is more general, since it can be used for statistical mixtures, but is it more fundamental than the ket for pure states? And, should we call these states at all? The notion of a zero entropy density operator is effectively equivalent to a projective notion of a (pure) quantum state, as Penrose noticed. Therefore, one might take the position that such density operators appropriately describe a ñphysical quantum stateî, taking the perspective that only that which has an impact in measurement problems can be considered to be physical. ñ(.83) I feel uncomfortable about regarding such a ïpure-state density matrixÍ as the appropriate mathematical representation of a ïphysical stateÍ. The phase factor (.83) is only ïunobservableÍ if the state under consideration represents the entire object of interest. When considering some state as part of a larger system, it is important to keep track of these phases (.83)î PenroseÍs main issue is related to the superposition principle. As Penrose puts it, the basic quantum linearity is obscured in the density operator description. Indeed, an objection of Penrose against the density operator is that the density operator makes complex the simpler linearity of the ket formalism. So far, PenroseÍs arguments equally apply to the density operator and to the density matrix. In pages 797 to 800 of Road to Reality, however, Penrose proceeds discussing what he considers to be the ñconfused ontological status of the density matrixî, in this case, the argument centers itself in the matrix and not in the operator. Indeed, some of the statements, used as counter-argumens apply correctly to the density matrix but not to the operator. The major argument refers to the inability of the density matrix to distinguish between different kinds of entangled pairs. For instance, consider the following scheme/example: the formulas] Even though we have two different kinds of entangled pairs, the density matrix is the same, that is, the density matrix does not seem to distinguish the bases. However, this is not the case if we take into account the density operator. The density operators are, not only, different, but if we determine the projection, for instance, of the second density operator with respect to the basis in which the first is represented, we obtain: the formulas] Indeed, the second operator is a statistical mixture between two pure states of superposition of 0> and 1> (+> and ->). What the above results show is that the projections of the second density operator to the basis {0>, 1>} and to the basis {+>,->}, differ, with respect to the probabilities assigned to the quantum events formalized by these projections. We can use a density matrix for reading probabilities, however, one must never confuse the matrix with the operator, and one must always use the operator, for the fundamental description. The problems of ontological confusion, raised by Penrose, can be raised with respect to the density matrix but not with respect to the density operator. This stresses the importance of precision of language, and the issue of the generalized practice of calling the density operator a density matrix (a practice followed by Penrose, and called into attention by Feynman in his Lectures on Physics, as a practical but mathematically imprecise simplification). This must be considered as a simplification of language, as Feynman stressed, but, nonetheless, it is a mathematical imprecision in the usage of the terminology, and, when dealing with fundamental arguments, one must take into account the distinction between the density matrix and the density operator. However, PenroseÍs argument about the phase seems to stand for both ñUnder normal circumstances, moreover, one must regard the density matrix as some kind of approximation to the whole quantum truth. For there is no general principle providing an absolute bar to extracting detailed information from the environment. Maybe a future technology could provide means whereby quantum phase relations can be monitored in detail, under circumstances where present-day technology would simply ïgive upÍ. It would seem that the resort to a density-matrix description is a technology-dependent prescription! With better technology, the state-vector could be maintained for longer, and the resort to a density matrix put off until things get really hopelessly messy! It would seem to be a strange view of physical reality to regard it to be ïreallyÍ described by a density matrix (.83)î Although these arguments may seem compelling, one may place a question regarding the statement on the approximation to the whole quantum truth, the question is: what about the so-called 'impure states'? As Penrose notices, one cannot discard that, at the quantum level, detailed phase relations may get ñlostî, because of some deep overriding basic principle. It is still too soon to discard such a hypothesis, and this, indeed, may be likely, if one considers a foamy Furthermore, there is still a division in the community in what including, more recently, Hawking (http://arxiv.org/abs/hep-th/ 0507171), that information may not be lost, we cannot yet reject this possibility. It seems that, accepting Penrose's argument, leads to the position that if we wish to use a fundamental mathematical description of physical reality, we must use two different formalisms, a ket for the pure states and a density operator for all the other cases, and we cannot discard the need for the usage of the density operator. Thus, to the first part of our main question (what is the most fundamental mathematical structure that should be used to describe the quantum system?)The arguments seem to point towards using the density operator only when necessary, as a technological tool. But is this sustainable? Do the phases matter? The answers to these questions cannot be entirely solved by appealing to mathematics alone. Indeed, a mathematician might be divided between: (a) a choice where one would work with what can be argued to be a more fundamental structure with respect to the information conserved in the description (the phase information), but two formalisms would be used for two different situations (pure vs impure states); (b) a choice where one works with a single formalism but part of the information (the phase) is lost. Since we are dealing with physics, all that matters is whether or not the phase is physically relevant, or, even, whether or not the density operator expresses, formally, the most fundamental physical nature, the normalized ket just being a useful representation, that can be shown to be equivalent to the density operator up to a global phase factor. In effect, so far, all that we can get from the system is the information contained in the density operator. The question of whether or not there might be some technology to recover the phase from a measurement, is still open to discussion. One may argue that, physically, the phase is irrelevant, one may alternatively argue that the phase is not physically irrelevant. However, to do the latter would demand the mathematical formulation of what might constitute a measurement procedure for the phase, leading inevitably to the problem of the physical meaning and measurability of a complex number. If one chooses to spend some time with this issue, one is led to this bifurcation of perspectives, where the choice depends less on mathematics and more on physics, in particular, our main question .82what is the most fundamental mathematical structure that should be used to describe the quantum system, and what is the nature of the physical semantics that this structure formalizes?é should be considered as a whole, since one cannot really consider the formalism, independently from the object of intentionality of the formalization (that which the formalization is about and that justifies the development of the formalization itself). What is fundamental for the mathematical structures of the formalism, may not be so for the object of formalization. In the end, the interpretation of quantum mechanics that one follows may decide the choice between the two paths, if one wishes to make such a choice at all, or if, and until, a fundamental thinking about physics demands such a choice. An interpretation of quantum mechanics that thinks about the nature of quantum processes, inevitably restricts our choices about what is fundamental due to the ontological and epistemological commitments that we assume, along the way of the construction of a scientifically grounded interpretation. In the interpretations that assign a physical nature to the wave function as corresponding to a pilot wave, the phases are relevant, even if they cannot be measured, since the fundamental object of formalization is that pilot wave. For a follower of Bohr, on the other hand, the whole discussion would be pointless, since the quantum formalism is just a useful tool used to predict results of experiments, whether we use a ket, a wave function or a density operator is irrelevant. Furthermore, Bohr was ñsuspiciousî of complex numbers, these could be useful tools, but, in the end, all that mattered were the predictions, and if a phase is unobservable by current technology it is a waste of time to think about it or to assign it a physical significance. In the Aristotle-based realist interpretation, followed by Heisenberg, the density operator should be taken as the formalization of the fundamental physical structure, since what the formalism ñformalizesî is the tendency of a potential alternative to be actualized, this intensity of the dynamis corresponds is quantifiable in terms of a degree, a degree with which probabilities coincide numerically, when these probabilities are interpreted as being proportional to the physical propensity of the potential alternative to be actualized, which is nothing but the intensity of the dynamis associated with that alternative. Taking this into account, the diagonal terms of the density operator are the fundamental structure, since they are in the direct correspondence with the object of formalization of the theory, i.e., they formalize the most fundamental physical structure, and their interpretation is naturally processual, a processual nature that is obscured by the ket representation. A closely related mathematical argument can be found in Bohm, Davies and Hiley's paper Algebraic Quantum Mechanics and Pregeometry (http:// arxiv.org/abs/quant-ph/0612002), where the authors built quantum theory from the primitive idempotents that are directly related to the different entries of the density operator. Bohm et al. show that the ket notation hides the fact that each ket represents an object with two labels. Thus, in the end, oneÍs solution to the phase problem and the answer to the central question placed here, depends on oneÍs choice of interpretation of quantum mechanics. === Subject: Re: On Penrose's argument against density operators posting-account=EI6PUAoAAAAazrTcDkROhitOfD8_tWFT 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > Why not email him I will do that. All the best, C. Pedro >http://cmathphil.blogspot.com/2008/07/on-penroses-argument-against-de... the usage of the density operator when dealing with pure quantum > states. The main arguments of Penrose are reviewed, and the reflection > proceeds around the main question: What is the most fundamental > mathematical structure that should be used to describe the quantum > system, and, what is the nature of the physical semantics that this > structure formalizes? [..] I believe Carl Brannen [http://carlbrannen.wordpress.com/] is also > enthusiastic about the density formulation of QM. Why not email him > John R Ramsden === Subject: sum of abs eigenvalues in terms of matrix elements? posting-account=WzP9FgoAAAANyEt4wx0YVvhakkQXYd72 Gecko/20061219 Fedora/1.5.0.9-1.fc6 Firefox/1.5.0.9 pango-text,gzip(gfe),gzip(gfe) We know some functions of eigenvalues of a matrix A have nice forms in terms of elements of the matrix. Let's denote eigenvalues by v_k and the element of A at i'th row and j'th column by a(i,j). We know for example about sum of eigen values sum_k (v_k) = sum_i a(i,i) Another example about sum of squared eigenvalues sum_i ((v_k)^2) = sum_i sum_j (a(i,j)^2) Now is there any way to express sum_k (abs(v_k)) in terms of elements of a? If not, are you aware of any good upper bound on this quanitity in terms of elements of a? G.D. === Subject: Re: sum of abs eigenvalues in terms of matrix elements? We know some functions of eigenvalues of a matrix A have nice forms in > terms of elements of the matrix. Let's denote eigenvalues by v_k and > the element of A at i'th row and j'th column by a(i,j). We know for example about sum of eigen values > sum_k (v_k) = sum_i a(i,i) > Another example about sum of squared eigenvalues > sum_i ((v_k)^2) = sum_i sum_j (a(i,j)^2) Now is there any way to express sum_k (abs(v_k)) in terms of elements > of a? No simple way, I think (except in cases where all v_k are positive or all are negative). > If not, are you aware of any good upper bound on this quanitity in > terms of elements of a? The largest absolute value of an eigenvalue is the spectral radius, which is bounded above by any matrix norm induced by a norm on vectors. And then the sum of the absolute values is bounded by n times this bound (where the matrix is n x n). So for example the sum of absolute values of eigenvalues is bounded by n max_i sum_j |a(i,j)|. This inequality is an equality in the case of the identity matrix, so in that sense it is best possible. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: #609 AP-coordinate system is a Digital System whereas the old Cartesian was Analog; new textbook: Mathematical Physics (AP-adic Primer) Let me try the previous post again to get the typing format to fit all the matrix points on the same line. They form a square Matrix, but the carriage format would confuse. Let me describe how to set up the AP-Coordinate System in 2D Euclidean, 2D Elliptic and 2D Hyperbolic All three of these are Digital Coordinate Systems whereas the old Cartesian is an Analog system. 2D Euclidean . . . . +0.000....00002 -0.0000...00002 +0.0000...00001, -0.0...001,+0.0...001,-0.0...002,+0.0...002,.. 0 The 0 sort of sticks out like a sore thumb but all the rest of the Reals, both positive and negative form a huge matrix. Now a graph of a function in 2D Euclidean Geometry would be that matrix above 2D Elliptic without the radix pi 9999...99999 . . . 000....00003 0000...00002 0000...00001, 000...0001 ,000...0002,000...0003, ......,999...999, pi 2pi 2D Elliptic with the radix pi 9999...99999r . . . 000....00003r 0000...00002r 0000...00001r, r, 00...001r,00...002r,00...003r,...., 999...999r, pi 2pi 2D Hyperbolic without the radix pi or e as imaginary -9999...99999 . . . -000....00003 -0000...00002 -0000...00001, -00...001 , -00...002, -00...003,..... ,-999...999, pi 2pi and or e as imaginary points Major difference between Elliptic and Hyperbolic is that Elliptic is all positive numbers and forms the numbers on a sphere surface whereas the Hyperbolic are all negative numbers on a pseudosphere surface. All of these three geometries are Digital since there is no continuity in any one of them. What makes Elliptic and Hyperbolic so bent and curved in space is the fact they do not have a negative sandwiched in between every positive as the Reals do and thus straighten out or flatten out the geometry. So to graph a function in any one of the three geometries of 2D, we simply plot the points in the matrix Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: #610 trouble with the Cartesian coordinate system is it never had any rivals, until now; new textbook: Mathematical Physics (AP-adic Primer) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > I wonder why the Descartes coordinate system ever became named > Cartesian. A dictionary > says Cartesius is a Latin form of Descartes. Maybe in past times, > places and names had > Latin versions. Renatus Cartesius is the latinized name, he was French. It was very common to have such latinized names because, since at least the middle ages, all that was doctus or otherwise sacred was writen and spoken in Latin. > Now my changes of the coordinate system was brought about not due to > continuity, but > keeping in mind that continuity brought me to this sandwiching of > negatives inbetween positives. I find your system very interesting. I hope I'll understand it someday! -LV === === Subject: Re: Short Mars travel times at high speed. really; line-of-sight is not the quickest path, nor the least-energy path. if you're ever going to find out what the shape of the box is (then, whether ye be in or out of it), you're going to have to at least look up the problem of the tautachrone, which is the same as the brachistoschrone, which is how Leibniz popularized his calculus (solved by Bernoulli, as well). but, please, don't googolplex it, or you'll never get it! a related, though maybe ill-posed problem, is, waht is the slowest path? AKA geometric optics or ray-tracing) is just an application of Fermat's least-time principle for light, or rather, Snell's extension of it to refraction. > No. æThe shortest distance may be a straight line, but > it's not always the most energy efficient. æRemember, thus: where is *what* ice, there? newsflash, icebergs calve off of icesheets, either because a) the icesheet is collapsing, or b) precipitation is greater. c) both of the above in some spots (the part of Antarctica that sticks-out into the weather e.g.: the Larsen-B iceshelf broke off, and the sky is glowing -- help!) I read that there are about 4x more polar bears, now, than there were a few decades ago, presumably because they like garbage as much as most other bears -- those feelthee Eskimos! > when all will seem strange and we will wonder where the ice is. > Why are not we able to connect the environmental dots? thus: yes, well, the daily rotation isn't a rational subdivisor, apparently, of the yearly revolution (at least, not a direct, linear relationship to be seen -- and why should there be?). the daylength can always be defined as 24 hours, regardless of the varying speed of rotation; Verdi's classical tuning is middle C = 256 cycles per second; A = 440Hz was established by the Nazis, at a concert in London, 'cause it's more brilliant for the times! thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the still-damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > instantaneous travel. thus: his statement could refer to raw coal rocks; I'm sure that there is a tremendous variation in the darkness of seams of coal! > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques,' or it's all ultimately geometry, anyway, somehow, a la Plato or Bucky Fullerofit); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar soros pamph.html http://larouchepub.com/other/2008/3526save nations parasites.html http://larouchepub.com/other/2008/3526zim brit op.html > incorrect value and the reasoning behind it even though our > civilisation achieved the core principles of 24 hours/360 degrees many > centuries ago and we use those principles day in and day out,then we > no longer deserve the title of 'civilisation' regardless oif how > advanced our technological achievements are. thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the extremely damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > How does that work? Well if you entangle photons in a weird way to produce instantaneous > instantaneous travel. thus: excellent quantifications; BG will never reply to it, I guess, although his statement could refer to raw coal rocks; I'm sure, though, that there is a tremendous variation in the darkness of seams of coal! > For anyone who believes the moon is too dark to wash out stars - > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques'); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar soros pamph.html http://larouchepub.com/other/2008/3526save nations parasites.html http://larouchepub.com/other/2008/3526zim brit op.html === Subject: Character of the squared tensor of a G-module posting-account=iBOKwQoAAAAqu6jW8ZJg_DbCQGFkeUMd Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Suppose that G is a finite group and V is an irreducible representation, with character X. Consider the tensor product $V otimes V$. Suppose that this product is reducible. How can we prove that X must be real? === Subject: Re: Character of the squared tensor of a G-module > Suppose that G is a finite group and V is an irreducible > representation, with character X. Consider the tensor product $V > otimes V$. Suppose that this product is reducible. How can we prove > that X must be real? If chi(g) is the character of V then the character of V otimes V is chi(g)^2, which need not be real. The character of V* otimes V, where V* is the dual of V, is real. Note that a representation is not necessarily real (ie representable by real matrices) even if its character is real. === Subject: Re: Character of the squared tensor of a G-module posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp Gecko/20080628 SUSE/2.0.0.15-1.1 Firefox/2.0.0.15,gzip(gfe),gzip(gfe) > Suppose that G is a finite group and V is an irreducible > representation, with character X. Consider the tensor product $V > otimes V$. Suppose that this product is reducible. How can we prove > that X must be real? If chi(g) is the character of V then the character of V otimes V > is chi(g)^2, which need not be real. > The character of V* otimes V, where V* is the dual of V, is real. Note that a representation is not necessarily real > (ie representable by real matrices) even if its character is real. That's true, but it is not hard to see that the image of the representation corresponding to the module V* otimes V has an invariant symmetric bilinear form, which implies by a result of Frobenius-Schur that the representation can be realized over R. Incidentally, V* otimes V has a 1-dimensional submodule, so it is always reducible when dim V > 1. Derek Holt. === Subject: Re: Character of the squared tensor of a G-module posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Suppose that G is a finite group and V is an irreducible > representation, with character X. Consider the tensor product $V > otimes V$. Suppose that this product is reducible. How can we prove > that X must be real? With great difficulty... Let us try an example, using GAP. Consider the alternating group of degree 5: gap> g := AlternatingGroup(5); Alt( [ 1 .. 5 ] ) and let us have GAP compute its character table for us: gap> tbl := CharacterTable(g); CharacterTable( Alt( [ 1 .. 5 ] ) ) gap> Display(tbl); CT1 2 2 2 . . . 3 1 . 1 . . 5 1 . . 1 1 1a 2a 3a 5a 5b 2P 1a 1a 3a 5b 5a 3P 1a 2a 1a 5b 5a 5P 1a 2a 3a 1a 1a X.1 1 1 1 1 1 X.2 3 -1 . A *A X.3 3 -1 . *A A X.4 4 . 1 -1 -1 X.5 5 1 -1 . . A = -E(5)-E(5)^4 = (1-ER(5))/2 = -b5 We see that the second character in the table, which is of degree 3, has non-real values. Let us call it chi: gap> chi := Irr(tbl)[2]; Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 3, -1, 0, -E(5)- E(5)^4, -E(5)^2-E(5)^3 ] ) The character of the tensor square of the module corresponding to chi is: gap> square := chi * chi; Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 9, 1, 0, -2*E(5)- E(5)^2-E(5)^3-2*E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 ] ) This square is a reducible (that's obvious, really, because it has degree 9...): gap> IsIrreducibleCharacter(square); false But as we can see, the squared character is not real. -- m === Subject: Re: The Einstein Hoax, GPS, and the soul - the Shortened Lunch Break - {HRI note 20080705-II} posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) these are largely a priori assumptions, that the Hubbleshift is the same as a Doppler shift, whereas the main effect of greater redshift with distance could simply be tied to the effects of vacuum; it's only relative! the anamolous associations of the quasars have been amply documented by the late Halton Arp, who was progressively denied access to time on the scopes, because of his seeming sacrilege. so, if the quasars are not as far away as a redshift-is-all-Dopplergangs, then they are not quite as powerful as calculated. anyway, I believe that one of the main interpretations is that quasars are associated with the births of new galaxies from older ones, or some thing, or it's meitotic, or what ever. as for the cosmic background radiation, I just read a snippet in *Harper's* that coincides with what I thought, that I'd read some when way back: ....Geneticists calculated taht humans nearly died out 70,000 years ago. Cosmologists postulated that the universe began not with a bang, but a splat, and suggested that the solar system may be saturated within a billion-light-year-wide bubble of low density, surrounded by a shell of high density, which would create, for observers on Earth, the illusion that the universe's expansion is accelerating. Geologists remain uncertain why the Earth hums. (Findings, July 2008) or, what I was originally thinking, is that COBE really saw what could not be known to be not just a local effect, anyway; the 2.7K is associatedd with what spectrum? anyway, photons are also created by antimatter; they are not antiphotons that anhialate the good vibrations o'light! > consists of a black hole which is actively moving matter and a galaxy. > Spectra have been taken of these objects so we know their red shift. > The distances are certainly cosmological. If they are NOT far away > what is the alternative explanation for red shift? All the photons being generated by matter. I don't know what you > mean. Photons must have a reason for their existence. Are you > referring to the 2.7K radiation? We know that after 300,000 years the > Universe became transparant. thus: strategically, it was Churchill who needlessly prolonged the war, and you can look that up on larouchepub.com; I forgot. the real tragedy, abundantly documented alsothereat, was Truman's going over MacArthur *and* Eisenhower's heads, to begin the nuclear age with an act of terrorism (although the firebombings were numerically worse). no aether is just an interpretation, plainly acknowledged by AE -- somebody on one of these fora recently quoted him on that. if you don't comprehend the proportionality of matter with energy, via c, you really have no gripe with the twin paradox ... what if you were there triplet deployed at 120-degree trajectory, away from them? > Why? There is no reference to any medium - material or otherwise - in > Maxwell's equations. > Lorentz transforms are only valid for constant velocity, and the > traveling twin accelerates when he turns around. Why can you not > understand this? thus: really; line-of-sight is not the quickest path, nor the least-energy path. if you're ever going to find out what the shape of the box is (then, whether ye be in or out of it), you're going to have to at least look up the problem of the tautachrone, which is the same as the brachistoschrone, which is how Leibniz popularized his calculus (solved by Bernoulli, as well). but, please, don't googolplex it, or you'll never get it! a related, though maybe ill-posed problem, is, waht is the slowest path? AKA geometric optics or ray-tracing) is just an application of Fermat's least-time principle for light, or rather, Snell's extension of it to refraction. > No. The shortest distance may be a straight line, but > it's not always the most energy efficient. Remember, thus: where is *what* ice, there? newsflash, icebergs calve off of icesheets, either because a) the icesheet is collapsing, or b) precipitation is greater. c) both of the above in some spots (the part of Antarctica that sticks-out into the weather e.g.: the Larsen-B iceshelf broke off, and the sky is glowing -- help!) I read that there are about 4x more polar bears, now, than there were a few decades ago, presumably because they like garbage as much as most other bears -- those feelthee Eskimos! > when all will seem strange and we will wonder where the ice is. > Why are not we able to connect the environmental dots? thus: yes, well, the daily rotation isn't a rational subdivisor, apparently, of the yearly revolution (at least, not a direct, linear relationship to be seen -- and why should there be?). the daylength can always be defined as 24 hours, regardless of the varying speed of rotation; Verdi's classical tuning is middle C = 256 cycles per second; A = 440Hz was established by the Nazis, at a concert in London, 'cause it's more brilliant for the times! thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the still-damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > instantaneous travel. thus: his statement could refer to raw coal rocks; I'm sure that there is a tremendous variation in the darkness of seams of coal! > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques,' or it's all ultimately geometry, anyway, somehow, a la Plato or Bucky Fullerofit); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar_soros_pamph.html http://larouchepub.com/other/2008/3526save_nations_parasites.html http://larouchepub.com/other/2008/3526zim_brit_op.html > incorrect value and the reasoning behind it even though our > civilisation achieved the core principles of 24 hours/360 degrees many > centuries ago and we use those principles day in and day out,then we > no longer deserve the title of 'civilisation' regardless oif how > advanced our technological achievements are. thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the extremely damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > How does that work? Well if you entangle photons in a weird way to produce instantaneous > instantaneous travel. thus: excellent quantifications; BG will never reply to it, I guess, although his statement could refer to raw coal rocks; I'm sure, though, that there is a tremendous variation in the darkness of seams of coal! > For anyone who believes the moon is too dark to wash out stars - > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques'); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar_soros_pamph.html http://larouchepub.com/other/2008/3526save_nations_parasites.html http://larouchepub.com/other/2008/3526zim_brit_op.html === Subject: Re: The Einstein Hoax, GPS, and the soul - the Shortened Lunch Break - {HRI note 20080705-II} posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) strategically, it was Churchill who needlessly prolonged the war, and you can look that up on larouchepub.com; I forgot. the real tragedy, abundantly documented alsothereat, was Truman's going over MacArthur *and* Eisenhower's heads, to begin the nuclear age with an act of terrorism (although the firebombings were numerically worse). no aether is just an interpretation, plainly acknowledged by AE -- somebody on one of these fora recently quoted him on that. if you don't comprehend the proportionality of matter with energy, via c, you really have no gripe with the twin paradox ... what if you were there triplet deployed at 120-degree trajectory, away from them? > Why? There is no reference to any medium - material or otherwise - in > Maxwell's equations. > Lorentz transforms are only valid for constant velocity, and the > traveling twin accelerates when he turns around. Why can you not > understand this? thus: really; line-of-sight is not the quickest path, nor the least-energy path. if you're ever going to find out what the shape of the box is (then, whether ye be in or out of it), you're going to have to at least look up the problem of the tautachrone, which is the same as the brachistoschrone, which is how Leibniz popularized his calculus (solved by Bernoulli, as well). but, please, don't googolplex it, or you'll never get it! a related, though maybe ill-posed problem, is, waht is the slowest path? AKA geometric optics or ray-tracing) is just an application of Fermat's least-time principle for light, or rather, Snell's extension of it to refraction. > No. The shortest distance may be a straight line, but > it's not always the most energy efficient. Remember, thus: where is *what* ice, there? newsflash, icebergs calve off of icesheets, either because a) the icesheet is collapsing, or b) precipitation is greater. c) both of the above in some spots (the part of Antarctica that sticks-out into the weather e.g.: the Larsen-B iceshelf broke off, and the sky is glowing -- help!) I read that there are about 4x more polar bears, now, than there were a few decades ago, presumably because they like garbage as much as most other bears -- those feelthee Eskimos! > when all will seem strange and we will wonder where the ice is. > Why are not we able to connect the environmental dots? thus: yes, well, the daily rotation isn't a rational subdivisor, apparently, of the yearly revolution (at least, not a direct, linear relationship to be seen -- and why should there be?). the daylength can always be defined as 24 hours, regardless of the varying speed of rotation; Verdi's classical tuning is middle C = 256 cycles per second; A = 440Hz was established by the Nazis, at a concert in London, 'cause it's more brilliant for the times! thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the still-damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > instantaneous travel. thus: his statement could refer to raw coal rocks; I'm sure that there is a tremendous variation in the darkness of seams of coal! > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques,' or it's all ultimately geometry, anyway, somehow, a la Plato or Bucky Fullerofit); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar_soros_pamph.html http://larouchepub.com/other/2008/3526save_nations_parasites.html http://larouchepub.com/other/2008/3526zim_brit_op.html > incorrect value and the reasoning behind it even though our > civilisation achieved the core principles of 24 hours/360 degrees many > centuries ago and we use those principles day in and day out,then we > no longer deserve the title of 'civilisation' regardless oif how > advanced our technological achievements are. thus: garbage, unless you consider the Alain Aspect version of Schroedinger's undead cat (Erwin's little gedanken joke) viz-a-vu the EPR paradox, which is just an interpretation (that is, the Copenhagen Schoolers' mystical write-up of this experiment; the cat has been dead for a hundred years, so that you probably wouldn't be able to find the putrid remains of *that* particular ****, including the extremely damp litter). what possible phenomenon requires speeds over light, other than scifi authorships? > How does that work? Well if you entangle photons in a weird way to produce instantaneous > instantaneous travel. thus: excellent quantifications; BG will never reply to it, I guess, although his statement could refer to raw coal rocks; I'm sure, though, that there is a tremendous variation in the darkness of seams of coal! > For anyone who believes the moon is too dark to wash out stars - > compare the number of stars you can see on a clear moonless night to > the number of stars you can see on a night when there's a full moon. thus: conjuring scifi is silly; it has always been a haven of supersillyousness or outright spookery, as per flatland by the ridiculous A.A.Skwared, 4D BS from the British Pyschol.Society etc., and the latterday mongering of timespace -- the arbitrary spacialization of time by means of a diagram, which is supposed to be an *aid* to comprehension of phasespace, not a reified ideal of itself, per Minkowski's youthful exhuberance. (y'know, Minkowski's stuff is really mathematically worthwhile, otherwise ... no, please, don't, stop -- mathematics ?!?!) *a priori* assumptions of megalithic structures on Venus, like Hoagland's Balls on Mars foolishness, really requires an actual program o'space to investigate, since nature is capable of quite awesome geometries (or 'hype-D physiques'); however, that was shotdown with Kennedy & Nixon: just as with our nuclear energy, we are still using '50s technology from the planet Marduk (per ScientologyTM .-) > You can consider them multidimensional beings who exist through time > the clearly intelligent infrastructure that can be seen as rational > and existing/coexisting on Venus? thus: that which causes the matter of time slowing in acceleration, is really the same as matter being energy, somehow, via their proportionality with c, the speed of light; that's incredibly obvious, although I know of no school of quantum, that says, how many quants of light make a proton e.g. (and, since the photons come in all sizes, it's moot .-) in any case, this is one of the properties of light that was experimentally verified in the 19th cce, although it is carachteristically never even mentioned, in favor of one of the EinsteinHubbleGodot paradoxi/ doctrines of the Department of Einsteinmania/ the Musical Department! thus: superstringtheory at least gets rid of that point, from the get-go & without further a-do, even if it's not just a string, a one-dimensional object ... howsoever it is that matter bends space, as measured by Gauss for the government of France in the 19th cce, and experimentally adduced by the classical Greek geometers, light travels through this bent medium, which apparently also alters the shape of it -- no timespace utterances needed, it's so very, blatantly bended-up! Roswell is a big double-entendre from WW2, but you could see that those who embrace it could go no further. that is really the gist of the Lt.Col. Corso School of Roswellology, that virtually *all* of 20th cce science & technology [http://www.21stcenturysciencetech.com/] came out of Corso's very own dyspersal of the pile of crap that fell out of the sky, there, to some big corporations; he had to wait til he was at death's door, to reap the deal for the book. that is to say, humans are incapable of generating ideas ... or, it just applies to Americans! > Einstein was not a hoax, you are. There IS a Roswell hoax, there was --Seargent Barracks Soros McCheeny Pepper, Give jihad a chance in The Sudan, Rhodesia, and other former colonial moments -- Yahoo!TM; you're going to feel my computerized draft, NO RHODESIA SCHOLARS IN HARM'S WAY! http://larouchepub.com/lar/2008/3526lar_soros_pamph.html http://larouchepub.com/other/2008/3526save_nations_parasites.html http://larouchepub.com/other/2008/3526zim_brit_op.html === Subject: Still thinking about the exterior product: vectors vs. forms posting-account=mV9EXQoAAACmCMM9qg0N4eJlXyr2Z93U MathPlayer 2.10b; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Some treatements assume we are dealing with forms, which, if I understand things correctly, are (multi)linear functions on vectors. Other authors admit vectors to the party. Do we have to choose one or the other and stick with it, or can we mix and match? And so forth. Any comments on sorting this out would be welcome. General comment on sci.math: there appear to be some professionals around willing to answer questions. That's great, but it's too bad there aren't a few more grad students/advanced undergrads in the mix, to take some of the load off the pros, and to foment more discussion along the gee, I'm really confused about ... are you? line. On any given day, there is someone, somewhere, struggling or recently having struggled with just about any given subject matter. Only connect. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I know the axioms. None of the axioms and no set of axioms together > refer to actual infinity. There is not the least hint. Right! That's what we've been trying to get you to understand for a LONG time! ZFC does not mention 'actual' nor 'potential'. Rather ZFC has a predicate that we read as 'is infinite'. Moreover, in an officially rigorous formalization of ZFC, the predicate 'is infninite' may not even have the English word 'infinite' but rather would just be the next available 1-place predicate symbol at the point the definition of the predicate symbol is introduced. > Only > inflexibility and obsession of set theorists together with the > unjustified application of finite logic raise the wrong impression of > actual infinity, of the existence of aleph 0 and so on. A lot of set theorists do regard existence in some ontoloigically or metaphysically realist sense. But that is not required just to work in set theory. Meanwhile Ex x=aleph 0 is just a trivial theorem. And the theorem that there exist infinite sets is not trivial, but it does not require a realist view just to see that it is a theorem (even though its theoremhood is a trivial consequence of an axiom). MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <1l3384dbg0dq85616gqptufmoichlbqpsm@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > No, he doesn't. That is the whole point: We're defining and entire >> string, any part of which is meaningless on its own. Then that is some form of contextual definition ... No, not necessarily; or at least it's NOT MEANT do be a contextual > definition. Yes, I thought about it later and realized 'contextual' is not the right word there. Another self-correction: I said that for 'card' Suppes takes an informal route later formalize with the axiom of choice. Actually, that's closer to Enderton in his set theory book, where he promises (his word) to justify the usage after introducing the axiom of choice. Suppes, on the other hand, takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, which he throws out when he can prove it as a theorem using choice. (My own preference is to give the ordinary definition of 'card' but without the axiom of choice, and use the Fregean method in case of improper referring, and show what can be proven just from Z set theory plus the numeration theorem. Then, finally, add the replacement schema and the axiom of choice so that we have ZFC, which proves the numeration theorem.) MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> No, he doesn't. That is the whole point: We're defining and entire >> string, any part of which is meaningless on its own. Then that is some form of contextual definition ... No, not necessarily; or at least it's NOT MEANT do be a contextual > definition. Yes, I thought about it later and realized 'contextual' is not the > right word there. Another self-correction: I said that for 'card' Suppes takes an informal route later formalize > with the axiom of choice. Actually, that's closer to Enderton in his > set theory book, where he promises (his word) to justify the usage > after introducing the axiom of choice. Suppes, on the other hand, > takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, > which he throws out when he can prove it as a theorem using choice. I am not familiar with Suppes' definition of 'equinumerous', but I guess it might be something like: for sets x and y, 'x and y are equinumerous' means 'there is a bijection between x and y'. Which would make Suppes' definition of 'card(x) = card(y)' no more virtuous than my definition of 'Card(A) <= Card(B)' === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I said that for 'card' Suppes takes an informal route later formalize > with the axiom of choice. Actually, that's closer to Enderton in his > set theory book, where he promises (his word) to justify the usage > after introducing the axiom of choice. Suppes, on the other hand, > takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, > which he throws out when he can prove it as a theorem using choice. I am not familiar with Suppes' definition of 'equinumerous', but I guess > it might be something like: > æ æfor sets x and y, 'x and y are equinumerous' means > æ æ'there is a bijection between x and y'. Right. > Which would make Suppes' definition of 'card(x) = card(y)' no more > virtuous than my definition of 'Card(A) <= Card(B)' No, Suppes does not give it as a DEFINITION. Rather, as I said, he gives it as an AXIOM. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > I said that for 'card' Suppes takes an informal route later formalize > with the axiom of choice. Actually, that's closer to Enderton in his > set theory book, where he promises (his word) to justify the usage > after introducing the axiom of choice. Suppes, on the other hand, > takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, > which he throws out when he can prove it as a theorem using choice. I am not familiar with Suppes' definition of 'equinumerous', but I guess > it might be something like: > æ æfor sets x and y, 'x and y are equinumerous' means > æ æ'there is a bijection between x and y'. Right. Which would make Suppes' definition of 'card(x) = card(y)' no more > virtuous than my definition of 'Card(A) <= Card(B)' No, Suppes does not give it as a DEFINITION. Rather, as I said, he > gives it as an AXIOM. As it is not an axiom of every form of ZF, it is not a necessary property of ZF (there is no need in general to have any meaning for Card(A) in ZF. So while Suppes may call it an axiom, it behaves much more like a definition, as it merely provides a form, 'card(x) = card(y)', which may always be eliminated. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> No, Suppes does not give it as a DEFINITION. Rather, as I said, he >> gives it as an AXIOM. >> Please, Virgil, note the differences between AXIOMs (proper) and DEFINITIONs. (There are several restrictions which a proper definition has to satisfy. An axiom [proper] is not restricted _in this way_.) As it is not an axiom of every form of ZF, it is not a necessary > property of ZF (there is no need in general to have any meaning for > Card(A) in ZF. > Huh? But you know that /cardinal numbers/ are a fundamental part in set theory? :-o (Yes, right, in ZFC they are just certain sets - still we ARE interested to define them and deal with them in ZFC.) Please tell a set theorist that there's no need for cardinal numbers in ZF!!! :-) So while Suppes may call it an axiom, ... > No, he does not *call* it an axiom. It *IS* an axiom in his system. ... it behaves much more like a definition, ... > No, it doesn't. But it behaves much like an AXIOM. (Well, not a big surprise, since it is one.) as it merely provides a form, 'card(x) = card(y)' ... > Nope. It does not only provide a form, but STATES that the cardinal number of A (i.e. card(A)) is identical with the cardinal number of B (i.e. Card(B) iff there is a bijection between A and B. ... which may always be eliminated. > No. Since Card (or rather K) is a primitive term in Suppes' system, it cannot be eliminated. B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > No, Suppes does not give it as a DEFINITION. Rather, as I said, he >> gives it as an AXIOM. > Please, Virgil, note the differences between AXIOMs (proper) and > DEFINITIONs. (There are several restrictions which a proper definition > has to satisfy. An axiom [proper] is not restricted _in this way_.) > As it is not an axiom of every form of ZF, it is not a necessary > property of ZF (there is no need in general to have any meaning for > Card(A) in ZF. Huh? But you know that /cardinal numbers/ are a fundamental part in set > theory? :-o (Yes, right, in ZFC they are just certain sets - still we > ARE interested to define them and deal with them in ZFC.) Please tell a > set theorist that there's no need for cardinal numbers in ZF!!! :-) > So while Suppes may call it an axiom, ... No, he does not *call* it an axiom. It *IS* an axiom in his system. > ... it behaves much more like a definition, ... No, it doesn't. But it behaves much like an AXIOM. (Well, not a big > surprise, since it is one.) A critical quality of a definition is that the definiens is replaceable by the definiendum. A critical property of Card(A) = Card(b) <==> 'there is bijection between A and B' , is that 'Card(A) = Card(b)' can be replaced by 'there is bijection between A and B', just like in a definition. > as it merely provides a form, 'card(x) = card(y)' ... Nope. It does not only provide a form, but STATES that the cardinal > number of A (i.e. card(A)) is identical with the cardinal number of B > (i.e. Card(B) iff there is a bijection between A and B. While the notion of cardinality does occur in ZFC, there is no need to denote the cardinality of a set A, however defined, by 'Card(A). It is sometimes denoted by '|A|', for example. > ... which may always be eliminated. No. Since Card (or rather K) is a primitive term in Suppes' system, > it cannot be eliminated. It can be if one is not using Suppes' system. You seem to be claiming that Suppes' system is required. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I said that for 'card' Suppes takes an informal route later formalize > with the axiom of choice. Actually, that's closer to Enderton in his > set theory book, where he promises (his word) to justify the usage > after introducing the axiom of choice. Suppes, on the other hand, > takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, > which he throws out when he can prove it as a theorem using choice. I am not familiar with Suppes' definition of 'equinumerous', but I guess > it might be something like: > æ æfor sets x and y, 'x and y are equinumerous' means > æ æ'there is a bijection between x and y'. Right. Which would make Suppes' definition of 'card(x) = card(y)' no more > virtuous than my definition of 'Card(A) <= Card(B)' No, Suppes does not give it as a DEFINITION. Rather, as I said, he > gives it as an AXIOM. As it is not an axiom of every form of ZF, it is not a necessary > property of ZF (there is no need in general to have any meaning for > Card(A) in ZF. So while Suppes may call it an axiom, it behaves much more like a > definition, as it merely provides a form, 'card(x) = card(y)', which may > always be eliminated. No, he states it explicitly as an axiom. And he DOES allow card(x) to also appear in other contexts aside from card(x) = card(y). That is his POINT in making it an axiom. So it is NOT a definition of a whole substring, as with your method, but rather indeed, just as he says, an axiom. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) I said that for 'card' Suppes takes an informal route later formalize > with the axiom of choice. Actually, that's closer to Enderton in his > set theory book, where he promises (his word) to justify the usage > after introducing the axiom of choice. Suppes, on the other hand, > takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, > which he throws out when he can prove it as a theorem using choice. I am not familiar with Suppes' definition of 'equinumerous', but I guess > it might be something like: > æ æfor sets x and y, 'x and y are equinumerous' means > æ æ'there is a bijection between x and y'. Right. Which would make Suppes' definition of 'card(x) = card(y)' no more > virtuous than my definition of 'Card(A) <= Card(B)' No, Suppes does not give it as a DEFINITION. Rather, as I said, he > gives it as an AXIOM. As it is not an axiom of every form of ZF, it is not a necessary > property of ZF (there is no need in general to have any meaning for > Card(A) in ZF. So while Suppes may call it an axiom, it behaves much more like a > definition, as it merely provides a form, 'card(x) = card(y)', which may > always be eliminated. No, he states it explicitly as an axiom. That does not make it a necessary axiom, as there are perfectly good axiom systems for ZF without it. And I didn't say it wasn't an axiom, I only said it behaved much like a definition, providing a way to eliminate expressions like 'card(x) = card(y)' wherever they appear. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> No, he states it explicitly as an axiom. > That does not make it a necessary axiom, ... > Who the hell claimed otherwise? :-o Of course it's not a necessary axiom, after all we can DEFINE the notion of Card in ZF(C), and then DERIVE Card(A) = Card(B) <-> A ~ B as a theorem. (Suppes introduces it _at an early stage of the development of the theory_ for PRACTICAL reasons, i.e. to facilitate certain derivations.) In my opinion he HAD to introduce it AS AN AXIOM, since he could not formulate it as a proper definition. ( Remember? ;-) as there are perfectly good axiom systems for ZF without it. > Sure, so what? In Chapter 4, a special axiom for cardinal numbers is presented and used mainly in the context of that chapter. This special axiom is not part of classical Zermelo-Fraenkel set theory, but it enormously facilitates the constructions of intuitive cardinal number theory within our axiomatic framework. (P. Suppes, Axiomatic Set Theory, p. 13) And I didn't say it wasn't an axiom, I only said it behaved much > like a definition, ... > No, it doesn't. providing a way to eliminate expressions like 'card(x) = card(y)' > wherever they appear. > Why should one like to do that? There is nothing to eliminate IF /card/ is a primitive notion of the system. B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > No, he states it explicitly as an axiom. > That does not make it a necessary axiom, ... Who the hell claimed otherwise? :-o Of course it's not a necessary axiom, after all we can DEFINE the > notion of Card in ZF(C), and then DERIVE Card(A) = Card(B) <-> A ~ B as a theorem. (Suppes introduces it _at an early stage of the > development of the theory_ for PRACTICAL reasons, i.e. to facilitate > certain derivations.) In my opinion he HAD to introduce it AS AN AXIOM, since he could not > formulate it as a proper definition. ( Remember? ;-) As I have not read Suppes, and do not intent to, as it seems to require all sorts of things things that are not necessary in other treatments, Suppes is not something that I will ever remember. > as there are perfectly good axiom systems for ZF without it. Sure, so what? In Chapter 4, a special axiom for cardinal numbers is presented and > used mainly in the context of that chapter. This special axiom is not > part of classical Zermelo-Fraenkel set theory, but it enormously > facilitates the constructions of intuitive cardinal number theory within > our axiomatic framework. (P. Suppes, Axiomatic Set Theory, p. 13) > And I didn't say it wasn't an axiom, I only said it behaved much > like a definition, ... No, it doesn't. Does it allow replacement of Card(A) = Card(B) wherever it occurs? If so, then it acts in that sense much like a definition. > providing a way to eliminate expressions like 'card(x) = card(y)' > wherever they appear. Why should one like to do that? It is not whether one would LIKE to or not. It is whether it is POSSIBLE to eliminate it, as one can always, at least in principle, do with definienda. There is nothing to eliminate IF /card/ > is a primitive notion of the system. > B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> I said that for 'card' Suppes takes an informal route later formalize >> with the axiom of choice. Actually, that's closer to Enderton in his >> set theory book, where he promises (his word) to justify the usage >> after introducing the axiom of choice. Suppes, on the other hand, >> takes card(x) = card(y) <-> x and y equinumerious as a formal axiom, >> which he throws out when he can prove it as a theorem using choice. > I am not familiar with Suppes' definition of 'equinumerous', but I guess > it might be something like: > for sets x and y, 'x and y are equinumerous' means > 'there is a bijection between x and y'. Which would make Suppes' definition of 'card(x) = card(y)' no more > virtuous than my definition of 'Card(A) <= Card(B)' > Huh?! What?! :-o That's just the STANDARD definition of 'equinumerous' - which is a proper one. Seems you still didn't get the point. A ~ B :<-> there is a bijection between A and B Nothing wrong with this definition. Suppes' definition of 'card(x) = card(y)' ... > ??? Suppes' _axiom_ (not _definition_!) concerning cardinality is: K(A) = K(B) <-> A ~ B, ruling the symbol K (which has to be added to the primitive symbols of our system). K(X) then is /the cardinal number of X/. As a _definition_ this would not be a proper one, but since it's an axiom there's no problem. Note that this way K(A) actually is an object; Suppes mentions in a footnote: --------------------------------- Note that independent of the axiom we may prove that (1) (Ex)(K(A) = x), once we admit the primitive term 'K', for it is a truth of logic that K(A) = K(A), and (1) is a logical consequence of this truth. --------------------------------- B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl > The first cardinal is zero for everyone who accepts that every finite > set has a cardinality. > > I do not disagree. But you just used the fact that the first (1st) > ordinal is 1 (or do you know of a 0th cardinal?). The first (1st) ordinal is 0. First is a common language term, 1 is a mathematical ordinal (and not a common language ordinal). -- 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: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > It may be an ordinal and a cardinal in the technical sense. It is not > a natural number. > But if you say that it is an ordinal then you should use it as an > ordinal, i.e., the zeroth ordinal is zero. > > But zeroth is *not* an ordinal number in the mathematical sense! > > Neither in any other. But first is *also* not an ordinal number in the mathematical sense! > It > could be an ordinal number in common language. Can't you see the > difference between 1-st and 1? > > As I told you so, you should know it. I said: > > Only in order to distinguish in what respect the natural is > used, the suffix is appended. You are wrong. In common language natural numbers are *not* ordinal numbers. Two is *not* an ordinal number, second is. And the two are not distinguished even by a suffix. > Of course the ordinal numbers can be > added: 1 + 3 = 4. And if you count to the first and starting again > from that one to the third, then you have counted in total to the > fourth. > > But that is cardinal addition, not ordinal addition. > > No that is ordinal addition. Cardinal addition shows that the first > cardinal and the third cardinal together are two cardinals (one of > them might become the pope). > > But apparently > first apple + second apple = third apple? And so, if you ask > somebody to pick up the first and second apple, you are happy if > he picks up the third apple? > > In some instances ordinal addition differs from cardinal addition. Set > theorist know that. So what is it. Are you happy if he picks up the third apple or not? -- 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: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > It may be an ordinal and a cardinal in the technical sense. It is not > a natural number. > But if you say that it is an ordinal then you should use it as an > ordinal, i.e., the zeroth ordinal is zero. But zeroth is *not* an ordinal number in the mathematical sense! Neither in any other. But first is *also* not an ordinal number in the mathematical sense! When you talk about a sequence, then you start with the least ordinal. When you talk mathematics, then you start denoting a sequence with the least ordinal in mathematics. If this is zero, then you should start with zeroth. It > could be an ordinal number in common language. Can't you see the > difference between 1-st and 1? As I told you so, you should know it. I said: Only in order to distinguish in what respect the natural is > used, the suffix is appended. You are wrong. In common language natural numbers are *not* ordinal > numbers. Two is *not* an ordinal number, second is. And the two are > not distinguished even by a suffix. I would say: 1 is the first ordinal number. 2 is the second ordinal number. first and second are adjectives that are formed using numbers. But I am not sure about that. I am not a linguist. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > But zeroth is *not* an ordinal number in the mathematical sense! > > Neither in any other. > > But first is *also* not an ordinal number in the mathematical sense! > > When you talk about a sequence, then you start with the least ordinal. Depends on what you use as index set. > When you talk mathematics, then you start denoting a sequence with the > least ordinal in mathematics. If this is zero, then you should start > with zeroth. Perhaps. Have a look at the sequence: a_i = 1/(i + 1), i >= 0 see? I started at 0. Whether indexing begins with 0 or with 1 or with whatever other number, depends on the index set you are using and the convention in use in that particular field. For Taylor series it is customary to start with 0. With vectors and matrices it is customary to start with 1. But both conventions have nothing to do with the ordinal numbers from mathematics. (And in a Taylor series, the first term is the constant term, aka a_0. And you can have MacLaurin series that can start with any negative integer as index.) > You are wrong. In common language natural numbers are *not* ordinal > numbers. Two is *not* an ordinal number, second is. And the two are > not distinguished even by a suffix. > > I would say: 1 is the first ordinal number. 2 is the second ordinal > number. first and second are adjectives that are formed using > numbers. But I am not sure about that. I am not a linguist. You are wrong about that. There is a similarity. But you are now using ordinal number in the terminology of common language. That is *not* the definition in mathematics. -- 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: A consideration concerning the diagonal argument of G. Cantor > When you talk about a sequence, then you start with the least ordinal. > When you talk mathematics, then you start denoting a sequence with the > least ordinal in mathematics. If this is zero, then you should start > with zeroth. The sequence 10,20,30,..., also has a first, second and third, so that by WM's argument the first ordinal is 10, the second 20, etc. WM conflates the value of a term in a sequence with the position of that term, thereby poisoning all his arguments about them. I would say: 1 is the first ordinal number. 2 is the second ordinal > number. Only in sequences beginning with 1 and 2 respectively, but there are uncountably many sequences that don't. All finite ordinals refer to positions in some sequence. The value at that position not necessarily an ordinal at all. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > A line is a heap. I only used this word and concept from >http://en.wikipedia.org/wiki/Heap_%28mathematics%29 > in contrast to the notion of set in order to prove formally that there > are aleph_0 symbols required to distinguish aleph_0 elements. > > What is the ternary operator? Moreover, I do not think that that kind > of heaps contains symbols, but objects and the same object does not occur > more than once in a heap. > > The heaps given there contain symbols, x and a iirc. There are no heaps shown on that page. You are confusing the ternary operator [.,.,.] with what you name heap. > I do not need > other properties than the fact that a multitude of similar symbols can > be contained in one and the same heap. Those heaps do not contain symbols but objects, and the objects are not repeated. As it is said there: a heap is an algebraic structture of a non-empty *set* H with a *ternary operation* denoted [x,y,z] in H which satisfies... So in principle a heap is a set and so does not contain duplicates. And clearly you are not able to even read such a page. > There is a perfectly good name for the things you call heap, and it is > not heap. > > That may be. But the name heap with the connotation of many or > multitude seems appropriate for my purpose. But it contradicts common mathematical use. multi-set is what you want andhas the advantage that it is the common term for that. > Second your notion of distinguishing appears a bit confused. You *can* > not distinguish things by symbols if the symbols themselves are not > distinguishable. > > That is an error. I mean you are in error. It is the fundament of > mathematics to count similar symbols by, for instance, So, say I have two symbols o and o, and two objects O and O. How do I distinguish the O's by the symbols o? I can attach an o to each O and so get O_o and O_o. How are they now distinguished? > You do not distinguish by *symbols*, but you distinguish > by the *number* of symbols in a heap. If heaps are different in > size (i.e. contain a different number of symbols), they are naturally > distinguished. As for each natural n, there is a heap with that number > of symbols (and so there are aleph_0 symbols), all the heaps are naturally > distinguished, and there are aleph_0 heaps and no heap has aleph_0 symbols. > > That is your old error. You could also claim that there is no natural > number with 10 symbols. > > As long as omega is understood as something that is completed, you > are wrong. If omega is not understood as something completed, then > Cantor's diagonal proof is wrong. There is no chance to circumvent > this self-contradiction. You consistently refuse to look at the arguments, or you do not understand the arguments. Cantor requires for his proof *only* that the diagonal is different for each element of the list. So given an element from the list it must be shown that the diagonal is different from it. Giving an element from the list is also giving the number of the element in the list (which is a natural number, say n), and the n-th digit of the diagonal is different from the n-th digit of that list element, and so they are different. That is all that is needed for the proof. Now, show *us* a proof that you if there are aleph_0 heaps there is necessarily a heap with aleph_0 symbols. > I used > it because by means of the pigeon-hole principle and by bijection for > all 4 cases (finite-finite, ..., infinite-infinite) there is an easy > and formal proof. > > What bijection? Do you mean from heaps to lines? But that is trivially > so the identity function works perfectly. And, as there is no infiniteth > line and no infiniteth heap, all is done. > > Here is my bijection for the last time: > > A--------> > B o > | oo > | ooo > v ... I see no bijection above. > For the minimal number A of symbols in at least one heap that is > required to distinguish a maximal number B of heaps in a set of heaps > we obtain following possibilities: > > 1) A = n in N <==> B = n in N. > 2) A = n in N <==> B > n in N. > 3) A > n in N <==> B = n in N. > 4) A > n in N <==> B > n in N. > > Case (1) is obviously correct for finite heaps and sets. Cases (2) and > (3) are impossible, because all natural numbers n are already absorbed > by case (1). There is nothing left but case (4). Because you *still* are not able to correctly spell out the bijection, this is nonsense. -- 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: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > That may be. But the name heap with the connotation of many or > multitude seems appropriate for my purpose. But it contradicts common mathematical use. multi-set is what you want > andhas the advantage that it is the common term for that. Multi-set has the disadvantage that its name contains the magic word set and so may lead to great confusion among set theorists. It is not uncommon that technical terms in mathematics have very different meanings. I use heap simply as a short hand for unary representations of a natural number. That means, my heaps contain only one sort of symbols. And I do not use multi-set because there is an empty multi- set but there is no empty heap. But if your aesthetic feelings cannot bear that term then use simply unary representations of naturals URNs. As there is onyl one type of symbol, we need not even colons within urns to distinguish the symbols. So our URNs are [o], [oo], [ooo], ... or, if different lines replace the brackets: o oo ooo ... Second your notion of distinguishing appears a bit confused. You *can* > not distinguish things by symbols if the symbols themselves are not > distinguishable. That is an error. I mean you are in error. It is the fundament of > mathematics to count similar symbols by, for instance, So, say I have two symbols o and o, and two objects O and O. How do > I distinguish the O's by the symbols o? I can attach an o to each O > and so get O_o and O_o. How are they now distinguished? You attach [o] to the first O and [oo] to the second. You consistently refuse to look at the arguments, or you do not understand > the arguments. Cantor requires for his proof *only* that the diagonal is > different for each element of the list. And he can claim and check that requirements for any finite initial segment of the list, not more. He cannot check it for the last elements, because there is none. All he can do is to require it for every initial segment. Just as I do. > So given an element from the list > it must be shown that the diagonal is different from it. Every element given from the list is an element of a finite initial segment. There is no difference with my approach. > Giving an element > from the list is also giving the number of the element in the list (which > is a natural number, say n), and the n-th digit of the diagonal is different > from the n-th digit of that list element, and so they are different. That > is all that is needed for the proof. That is the same for my proof. Given any n I can prove that this one and every other one is covered by my proof, simply because there is no n that is not covered by my proof. (Here I refer to my proof that the complete sequence of rational numbers can be inserted in the real line.) Now, show *us* a proof that you if there are aleph_0 heaps there is > necessarily a heap with aleph_0 symbols. Here it is: > 1) A = n in N <==> B = n in N. > 2) A = n in N <==> B > n in N. > 3) A > n in N <==> B = n in N. > 4) A > n in N <==> B > n in N. Case (1) is obviously correct for finite heaps and sets. Cases (2) and > (3) are impossible, because all natural numbers n are already absorbed > by case (1). There is nothing left but case (4). Because you *still* are not able to correctly spell out the bijection, > this is nonsense. For every finite number n there is no doubt that at least one heap with n symbols must be in a set that contains n distinct heaps. What sort of problems do you have with this fact? It is easily seen by complete induction. I used > it because by means of the pigeon-hole principle and by bijection for > all 4 cases (finite-finite, ..., infinite-infinite) there is an easy > and formal proof. What bijection? Do you mean from heaps to lines? But that is trivially > so the identity function works perfectly. And, as there is no infiniteth > line and no infiniteth heap, all is done. Here is my bijection for the last time: A-------- B o > | oo > | ooo > v ... I see no bijection above. Then look again. Hint: There is a diagonal of the fourth quadrant. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl > That may be. But the name heap with the connotation of many or > multitude seems appropriate for my purpose. > > But it contradicts common mathematical use. multi-set is what you want > andhas the advantage that it is the common term for that. > > Multi-set has the disadvantage that its name contains the magic word > set and so may lead to great confusion among set theorists. On the other hand it is well defined in set theory, so I have no idea why you think it may lead to great confusion. > It is not > uncommon that technical terms in mathematics have very different > meanings. Right. But that is not so frequent in a single field of mathematics. But the term heap had only a single meaning in mathematics, and that was *not* the meaning you intende. > I use heap simply as a short hand for unary representations > of a natural number. Of course, this is again imprecise. > That means, my heaps contain only one sort of > symbols. And I do not use multi-set because there is an empty multi- > set but there is no empty heap. But if your aesthetic feelings cannot > bear that term then use simply unary representations of naturals > URNs. As there is onyl one type of symbol, we need not even colons > within urns to distinguish the symbols. So our URNs are [o], [oo], > [ooo], ... or, if different lines replace the brackets: Ok. So you now use URN's which are heaps. > That is an error. I mean you are in error. It is the fundament of > mathematics to count similar symbols by, for instance, > > So, say I have two symbols o and o, and two objects O and O. How do > I distinguish the O's by the symbols o? I can attach an o to each O > and so get O_o and O_o. How are they now distinguished? > > You attach [o] to the first O and [oo] to the second. In that case I do not distinguish by symbols, but by heaps, and so the number of symbols is irrelevant and we need to talk only about the number of heaps. > You consistently refuse to look at the arguments, or you do not understand > the arguments. Cantor requires for his proof *only* that the diagonal is > different for each element of the list. > > And he can claim and check that requirements for any finite initial > segment of the list, not more. It is *not* about initial segments, it is about individual elements. And it does not need checking, it is proven. > All he can do is to require it for > every initial segment. Just as I do. He does not require it for initial segments, but for individual elements. That is something different. > So given an element from the list > it must be shown that the diagonal is different from it. > > Every element given from the list is an element of a finite initial > segment. There is no difference with my approach. There is, a fundamental difference. As I did show in my side by side presentation about Cantor's proof and you proof. > Giving an element > from the list is also giving the number of the element in the list (which > is a natural number, say n), and the n-th digit of the diagonal is > different from the n-th digit of that list element, and so they are > different. That is all that is needed for the proof. > > That is the same for my proof. Given any n I can prove that this one > and every other one is covered by my proof, simply because there is no > n that is not covered by my proof. That is unclear reasoning. What you *do* proof is that when you apply your operation for all elements upto element n, that what you state is true. This does *not* mean that it is valid when you apply your operation to *all* elements. > (Here I refer to my proof that the > complete sequence of rational numbers can be inserted in the real > line.) It can, but not doing so step by step. > Now, show *us* a proof that you if there are aleph_0 heaps there is > necessarily a heap with aleph_0 symbols. > > Here it is: > > 1) A = n in N <==> B = n in N. > 2) A = n in N <==> B > n in N. > 3) A > n in N <==> B = n in N. > 4) A > n in N <==> B > n in N. > > Case (1) is obviously correct for finite heaps and sets. Cases (2) and > (3) are impossible, because all natural numbers n are already absorbed > by case (1). There is nothing left but case (4). > > Because you *still* are not able to correctly spell out the bijection, > this is nonsense. > > For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. Yes, that is true. So we are in case (1). So why is there a need for a heap with aleph_0 symbols? And again, *what* is the bijection you are talking about? Not in pictures, but spell it out in words. The bijection f is from the set A to the set B, where for a in A, f(a) in B. What is A, what is B, what is f? > What bijection? Do you mean from heaps to lines? But that is > the heaps are lines, so the identity function works perfectly. > And, as there is no infiniteth line and no infiniteth heap, all > is done. > > Here is my bijection for the last time: > > A--------> > B o > | oo > | ooo > v ... > > I see no bijection above. > > Then look again. Hint: There is a diagonal of the fourth quadrant. A bijection is a mapping from a set A to a set B with particular properties. What is the set A, what is the set B? I assume the set B is the set of lines, is the set A the set of columns? Well, in that case for each n in N there is a line in B and a column in A, so there is a trivial bijection here, and no need for something with aleph_0 symbols. (Note: the n-th column starts at the end of the n-th line, so for each line there is a corresponding column.) -- 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: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It is not > uncommon that technical terms in mathematics have very different > meanings. Right. But that is not so frequent in a single field of mathematics. But > the term heap had only a single meaning in mathematics, and that was > *not* the meaning you intende. The term heap was used already in the Papyrus Rhind to denote a collection of similar objects. And in the present discussion there is no danger to mistake such a heap with an algebraic structure. > I use heap simply as a short hand for unary representations > of a natural number. Of course, this is again imprecise. Why do you believe that? That means, my heaps contain only one sort of > symbols. And I do not use multi-set because there is an empty multi- > set but there is no empty heap. But if your aesthetic feelings cannot > bear that term then use simply unary representations of naturals > URNs. As there is onyl one type of symbol, we need not even colons > within urns to distinguish the symbols. So our URNs are [o], [oo], > [ooo], ... or, if different lines replace the brackets: Ok. So you now use URN's which are heaps. URNs which are unary representations of naturals. That is the point, not how we call them. That is an error. I mean you are in error. It is the fundament of > mathematics to count similar symbols by, for instance, So, say I have two symbols o and o, and two objects O and O. How do > I distinguish the O's by the symbols o? I can attach an o to each O > and so get O_o and O_o. How are they now distinguished? You attach [o] to the first O and [oo] to the second. In that case I do not distinguish by symbols, but by heaps, and so the > number of symbols is irrelevant and we need to talk only about the number > of heaps. The number of symbols in a set of URNs is an upper bound for the number of URNs in a set of URNs. This holds for every finite number of symbols respectively URNs. Therefore it cannot get false for the number of URNs surpassing every finite number (in case such a number exists). > You consistently refuse to look at the arguments, or you do not understand > the arguments. Cantor requires for his proof *only* that the diagonal is > different for each element of the list. And he can claim and check that requirements for any finite initial > segment of the list, not more. It is *not* about initial segments, it is about individual elements. And > it does not need checking, it is proven. It is proven for every member of every initial segment. Either this implies the whole set or it does not. In any case the same holds for Cantor's proof as well as for mine. All he can do is to require it for > every initial segment. Just as I do. He does not require it for initial segments, but for individual elements. > That is something different. No. Using initial segments, as they are used in my proof, does not yields a weaker result, because for every individual element there is an initial segment. There is no individual element outside of all initial segments. So from the standpoint of logic, a proof using initial segments cannot be weaker than a proof using single elements. So given an element from the list > it must be shown that the diagonal is different from it. Every element given from the list is an element of a finite initial > segment. There is no difference with my approach. There is, a fundamental difference. As I did show in my side by side > presentation about Cantor's proof and you proof. You did nothing of that kind. Your desire is to believe so - that's all. Giving an element > from the list is also giving the number of the element in the list (which > is a natural number, say n), and the n-th digit of the diagonal is > different from the n-th digit of that list element, and so they are > different. That is all that is needed for the proof. That is the same for my proof. Given any n I can prove that this one > and every other one is covered by my proof, simply because there is no > n that is not covered by my proof. That is unclear reasoning. What you *do* proof is that when you apply your > operation for all elements upto element n, that what you state is true. > This does *not* mean that it is valid when you apply your operation to > *all* elements. Same does Cantor. (Here I refer to my proof that the > complete sequence of rational numbers can be inserted in the real > line.) It can, but not doing so step by step. That is the complete sequence! Doing something step by step without an end, means doing something for a complete infinite set. There is nothing more you can do. Now, show *us* a proof that you if there are aleph_0 heaps there is > necessarily a heap with aleph_0 symbols. Here it is: 1) A = n in N <==> B = n in N. > 2) A = n in N <==> B > n in N. > 3) A > n in N <==> B = n in N. > 4) A > n in N <==> B > n in N. Case (1) is obviously correct for finite heaps and sets. Cases (2) and > (3) are impossible, because all natural numbers n are already absorbed > by case (1). There is nothing left but case (4). Because you *still* are not able to correctly spell out the bijection, > this is nonsense. For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. Yes, that is true. So we are in case (1). So why is there a need for > a heap with aleph_0 symbols? And again, *what* is the bijection you > are talking about? Not in pictures, but spell it out in words. > The bijection f is from the set A to the set B, where for > a in A, f(a) in B. > What is A, what is B, what is f? A = set of symbols in URNs, call the elements n B = set of initial segments of natural numbers, call the elements f(n) The bijection is f(n) = n What bijection? Do you mean from heaps to lines? But that is > the heaps are lines, so the identity function works perfectly. > And, as there is no infiniteth line and no infiniteth heap, all > is done. Here is my bijection for the last time: A-------- B o > | oo > | ooo > v ... I see no bijection above. Then look again. Hint: There is a diagonal of the fourth quadrant. A bijection is a mapping from a set A to a set B with particular properties. > What is the set A, what is the set B? A = set of symbols in URNs, call the elements n B = set of initial segments of natural numbers, call the elements f(n) The bijection is f(n) = n Both sides are finite or both sides are actually infinite. > (Note: the n-th column > starts at the end of the n-th line, so for each line there is a corresponding > column.) As long as the initial segments of lines have ends (= are finite). And in none of these cases we have a number larger than every natural number. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. Look! Over There! A Pink Elephant! There is no doubt that at least one heap with aleph_0 symbols must be in a set that contains aleph_0 distinct heaps. - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. Look! Over There! A Pink Elephant! There is no doubt that at least one heap with aleph_0 symbols > must be in a set that contains aleph_0 distinct heaps. Not in ZF. The set of all finite multisets with only 'o' as member does not contain an infinite multiset. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > There is no doubt that at least one heap with aleph_0 symbols > must be in a set that contains aleph_0 distinct heaps. Not in ZF. The set of all finite multisets with only 'o' as member does > not contain an infinite multiset. because actually the set of all finite multisets is not actually infinte. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor There is no doubt that at least one heap with aleph_0 symbols > must be in a set that contains aleph_0 distinct heaps. Not in ZF. The set of all finite multisets with only 'o' as member does > not contain an infinite multiset. because actually the set of all finite multisets is not actually > infinte. Ordered sets, which that set is, which do not have last/largest members, which that set does not, are neither finite nor potentially infinite, by any test of either finiteness or potential infiniteness. And it passes the two basic tests for (actual) infiniteness in ZF: (1) it is nonempty and does not biject with any set of n members where n is a positive natural (equivalently, it surjects to every set of n elements, for n a natural). (2) there exist injections from it to proper subsets of it. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. Look! Over There! A Pink Elephant! There is no doubt that at least one heap with aleph_0 symbols > must be in a set that contains aleph_0 distinct heaps. Not in ZF. The set of all finite multisets with only 'o' as member does > not contain an infinite multiset. Stop thinking about things and look at the nice Pink Elephant. - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor That may be. But the name heap with the connotation of many or > multitude seems appropriate for my purpose. But it contradicts common mathematical use. multi-set is what you want > andhas the advantage that it is the common term for that. Multi-set has the disadvantage that its name contains the magic word > set and so may lead to great confusion among set theorists. Set theorists are a good deal less likely to be confused by describing what has all the set-theoretical properties of a multiset by the word 'multiset' than by describing it by the word 'heap' which has, mathematically,a meaning incompatible with what WM is trying to describe. > It is not > uncommon that technical terms in mathematics have very different > meanings. I use heap simply as a short hand for unary representations > of a natural number. Since in mathematical usage, heap already has another well-defined but incompatible meaning, but multiset has a meaning that precisely descibes what WM is trying to speak about, WM is being deliberately anti-mathematical. That means, my heaps contain only one sort of > symbols. And I do not use multi-set because there is an empty multi- > set but there is no empty heap. But if your aesthetic feelings cannot > bear that term then use simply unary representations of naturals > URNs. As there is onyl one type of symbol, we need not even colons > within urns to distinguish the symbols. So our URNs are [o], [oo], > [ooo], ... or, if different lines replace the brackets: > o > oo > ooo > ... Or, even more descriptively, <1,o> <2,o> <3,o> .... So, say I have two symbols o and o, and two objects O and O. How do > I distinguish the O's by the symbols o? I can attach an o to each O > and so get O_o and O_o. How are they now distinguished? You attach [o] to the first O and [oo] to the second. How can one tell which is first and which is second unless they are already distinguished? You consistently refuse to look at the arguments, or you do not understand > the arguments. Cantor requires for his proof *only* that the diagonal is > different for each element of the list. And he can claim and check that requirements for any finite initial > segment of the list, not more. Anyone can check it for each finitely positioned element of the list, and every element of any list is, by definition, finitely positioned. Or does WM claim that there must be some term in such a list that is infinitely positioned? > He cannot check it for the last > elements, because there is none. All he can do is to require it for > every initial segment. Just as I do. If one takes the last element of every finite initial segment of a list, as Cantor does, one has covered every element of that list. Unless WM's lists contain terms which are more than finitely far down the list of elements. > So given an element from the list > it must be shown that the diagonal is different from it. Every element given from the list is an element of a finite initial > segment. If one takes the last element of every finite initial segment of a list, as Cantor does, one has covered every element of that list. For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. But there is set of heaps for which there is no such n limiting the number of heaps in it. And for that set, WM's mytheology does not hold. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > Every element given from the list is an element of a finite initial > segment. If one takes the last element of every finite initial segment of a list, > as Cantor does, one has covered every element of that list. For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. But there is set of heaps for which there is no such n limiting the > number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that set, as Cantor does, one has covered every element of that set. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Every element given from the list is an element of a finite initial > segment. If one takes the last element of every finite initial segment of a list, > as Cantor does, one has covered every element of that list. For every finite number n there is no doubt that at least one heap > with n symbols must be in a set that contains n distinct heaps. But there is set of heaps for which there is no such n limiting the > number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that > set, > as Cantor does, one has covered every element of that set. > The point, which WM carefully avoids, is that there is no member of THAT set of pseudoheaps that represents the 'number' of pseudoheaps in it. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that > set, > as Cantor does, one has covered every element of that set. Proving that P is true for every *element* of a set may or may not prove that P is true for the set. - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that > set, > as Cantor does, one has covered every element of that set. Proving that P is true for every *element* of a set > may or may not prove that P is true for the set. For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that > set, > as Cantor does, one has covered every element of that set. Proving that P is true for every *element* of a set > may or may not prove that P is true for the set. For every line of Cantor's list it is true that this line does not > contain the diagonal number. Correct. Now step 2. To say that the diagonal number is in a list means that the diagonal number is equal to some element of the list. The fact that the diagonal element is not equal to any element of the infinite list means that the diagonal element is not in the infinite list. may or may not includes the case may. -William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Correct. Now step 2. To say that the diagonal > number is in a list means that > the diagonal number is equal to some element of > the list. The fact that the diagonal > element is not equal to any element of the infinite > list means that the diagonal element is not in the > infinite list. > Since WM suffer from a serious quantifier dyslexia (and a general inability to think mathematically) he won't get it. You will see! :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor number of heaps in it. And for that set, WM's mytheology does not hold. If one takes the last element of every finite initial segment of that > set, > as Cantor does, one has covered every element of that set. Proving that P is true for every *element* of a set > may or may not prove that P is true for the set. For every line of Cantor's list it is true that this line does not > contain the diagonal number. > Nevertheless the diagonal number may be in the infinite list. Which shell is WM hiding that particular pea under? As usual in such shell games, it is not under any shell but off up the manipulator's sleeve. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Proving that P is true for every *element* of a set > may or may not prove that P is true for the set. > Consider the set of all blue objects (which are not sets) in the world, we may reasonably doubt that this set is blue. :-) Or consider the set consisting of all objects (non-sets that is) with a mass of 1 kg. Now does this mean that this set has a mass of 1 kg? Note that WM never learned to comprehend such fine distinctions. The sad truth is that he's just mentally inapt to do that. A serious example: The set containing all natural numbers (as elements) is not a natural number itself. We certainly won't claim that the fact that any natural number is in N implies that N is in N too. (*sigh*) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor >> For every finite number n there is no doubt that at least one heap >> with n symbols must be in a set that contains n distinct heaps. > But there is set of heaps for which there is no such n limiting the > number of heaps in it. And for that set, WM's mytheology does not hold. > Or at least such a set is conceivable (outside crank-town that is). B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080325 Ubuntu/7.10 (gutsy) Firefox/2.0.0.13,gzip(gfe),gzip(gfe) > Here is my bijection for the last time: Ah! A Mueckenheim promise. Will it be kept? > A-------- B o > | oo > | ooo > v ... [Dik] I see no bijection above. Obviously, since WM is a mathematical ignoramus, he can't be expected to write anything in normal mathematical notation, but just draws nursery pictures. However, I think you lack imagination here, Dik... there is an obvious bijection suggested by these arrows. Replacing some of the 'o's by different symbols, and filling out a bit: > A-------- B 0 > | 11 > | 2o2 > | 3oo3 > | 4ooo4 > v ... Map the 0 to itself, and the left 1 to the right 1, and so on. Actually this simply says that in any list of two-ended objects, there is a canonical mapping from the set of left ends to the set of right ends. Right, where's the problem? None, if this is a one-ended list: it has a top end, shown above, and extends downward without ever ending, so there is no bottom edge. However, I think it's obvious that a combination of WM's exceedingly feeble grasp of reading with some highly sloppy comments from people who might know better (except that if they did they realise that arguing with cranks is a waste of time, and they've already gone somewhere else) leads WM to think that omega (or any other Greek letter) is the number at the bottom end of the list. Therefore, there is a last element in the list which looks like (w for omega): woooo .... [omega-2 'o's] ... oooow Hence all his confused contradictions etc. Hope (with zero confidence) that this helps... Brian Chandler === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Hence all his confused contradictions etc. Hope (with zero confidence) that this helps... > He once got crazy when he was told that there is no index w in the decimal expansion of a real number (i.e. in the sequence of it's digits). Well... He claimed (and still does, I think) that two real numbers x < 1 an y < 1 can be different even though for all i e IN : [x]_i = [y]_i. (Where [z]_i denotes the i-th decimal in the decimal expansion of z.*) B. *) Always using the one without infinitely many 9's at the end. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> More relaxed: >> card(y) = the least ordinal equinumerous with y. > Gotcha! That's why the i-Operator would be such a fine thing in a formal system. > Actually in standard math many definitions are formulated in this way, > i.e. using a definite description (which I consider an implicit or > hidden use of the i-Operator). > In particular, when we _define_ AuB to be the set of all x such that x is an element of A or x is an element of B we do need to show from the axioms that there _is_ such a set. Here Ullrich mentions the definition: A u B =df the set of all x such that x e A or x e B. or in symbols: A u B =df iy(Ax(x e y <-> x e A v x e B)). Using Frege's approach, we do not even need to show from the axioms that for any A, B there actually is exactly one set y such that Ax(x e y <-> x e A & x e B). Of course it's useful to prove at some later stage (as a theorem) that we actually have Ax(x e A u B <-> x e A v x e B) for any A, B. :-) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> That's why the i-Operator would be such a fine thing in a formal > system. > But how do you add such an operator to a classical logic? In the way Bernays did it in his book 'Axiomatic Set Theory'; a > variant of Frege's original approach. Ok, I just wondered what you had in mind; you use a default referent for i-terms that whose descriptions don't pick out a unique thing. I'll have to look at the version you mention that doesn't require introducing a special constant for the default referent, as that is something that has always put me off using an i-Operator. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <6r9284dm7tkgtoglffadpqbarnbj6abqgm@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) 8213 On Jul 20, 11:26 am, Chris Menzel variant of Frege's original approach. Ok, I just wondered what you had in mind; you use a default referent > for i-terms that whose descriptions don't pick out a unique thing. I'll > have to look at the version you mention that doesn't require introducing > a special constant for the default referent, as that is something that > has always put me off using an i-Operator. Interesting little pedantic speedbump here. The Fregean method, as usually applied, say in set theory, uses the fact that we can define at least one unique object (say, the empty set, as it usually is the one we turn to). I.e., we can conservatively extend with a 0-place function symbol. But what if the theory does not pick out a unique object? Well, (1) what interesting mathematical theory doesn't define at least one unique object? (2) Aside from (1) with its subjectivity of interesting, we could reformulate our notion of a language for a theory so that every language has at least one 0-place function symbol, and allow that the language may include the symbol in a vacuous way of not having any non-logical axioms about it; then we use it as the default constant for definitions that aren't otherwise set up with an existence and uniqueness theorem (or for all definitions of function symbols, for that matter). MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> But how do you add such an operator to a classical logic? > In the way Bernays did it in his book 'Axiomatic Set Theory'; a >> variant of Frege's original approach. > Ok, I just wondered what you had in mind; you use a default referent > for i-terms that whose descriptions don't pick out a unique thing. > I'll have to look at the version you mention [...]. > The idea is simple. Bernays introduces as a primitive of the logical system the operator i_t, where t is a (any) term. This operator is ruled by two axioms (here I'm deviating from Bernays for simplicity/clarity): A1 E!xPhi[x] -> Phi[i_txPhi[x]], A2 ~E!xPhi[x] -> i_txPhi[x] = t. So we do not need at this stage of the theory (i.e. at the level of first-order predicate logic with identity) a constant to fix the denotation if ixPhi[x] in case of ~E!xPhi[x]. Later, when we have, say 0 (or {}), at our disposal we may define ixPhi[x] =df i_0xPhi[x]. We then get T1 E!xPhi[x] -> Phi[ixPhi[x]], T2 ~E!xPhi[x] -> ixPhi[x] = 0. Just what we want. :-) ...that doesn't require introducing a special constant for the default > referent, as that is something that has always put me off using an i- > Operator. > I mean, we cannot avoid (in a classical framework) to give the term ixPhi[x] SOME denotation, even when it wouldn't ordinarily have one, i.e. in case of E!xPhi[x]. B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> ... something that has always put me off using an i-Operator. >> The funny thing is that many mathematicians use it freely in ordinary (non formal) lingo of math. For example they formulate things like: ... when we _define_ AuB to be the set of all x such that x is an element of A or x is an element of B we do need to show from the axioms that there _is_ such a set. (David C. Ullrich in a recent post) Well, actually, using Frege's approach and a formal theory of the i-Operator, we do not. ;-) Anyway, this definition can be formalized the following way: A u B =df iy(Ax(x e y <-> x e A v x e B)). Later we can show (from the axioms) that for any sets A and B there actually is exactly one set of all x such that x is an element of A or x is an element of B. In symbols: E!y(Ax(x e y <-> x e A v x e B)). And with this theorem (and the definition mentioned above) we get immediately from the axioms of the i-operator: Ax(x e A u B <-> x e A v x e B) for any A, B. Imho the i-Operator would (to a certain extend) just reflect normal mathematical practice (at a formal level). B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) ... something that has always put me off using an i-Operator. >> The funny thing is that many mathematicians use it freely in ordinary > (non formal) lingo of math. For example they formulate things like: ... when we _define_ AuB to be the set of all x such that > x is an element of A or x is an element of B we do need > to show from the axioms that there _is_ such a set. (David C. Ullrich in a recent post) Well, actually, using Frege's approach and a formal theory of the > i-Operator, we do not. ;-) Anyway, this definition can be formalized the following way: A u B =df iy(Ax(x e y <-> x e A v x e B)). Later we can show (from the axioms) that for any sets A and B there > actually is exactly one set of all x such that x is an element of A or x > is an element of B. In symbols: E!y(Ax(x e y <-> x e A v x e B)). And with this theorem (and the definition mentioned above) we get > immediately from the axioms of the i-operator: Ax(x e A u B <-> x e A v x e B) for any A, B. Imho the i-Operator would (to a certain extend) just reflect normal > mathematical practice (at a formal level). > B. > The i-operator ix phi, the (unique) x such that phi, is used in Bell/Machover, _A Course in Mathematical Logic_ (1977), as a way to introduce what they call virtual terms (p. 106) in logic. Later, in set theory (p. 465), they define the set builder as {x : phi} =df iz(Ax(x e z <-> phi)) where z is not free in phi. So, for example, the union of two sets could be equivalently defined as either A u B = iz(Ax(x e z <-> x e A v x e B)) or A u B = {x : x e A v x e B} For set theory definitions, they usually resort to the latter since it is more compact, but occasionally they revert to the i-operator where it is more expedient such as in the definition of function value. Note that their set builder definition doesn't allow expressing proper classes, since it would evaluate to the empty set in such a case. (In their entire ZFC development, they avoid speaking of proper classes by using wff metavariables in place of the more typical class variables of other authors. Suppes does this also, although B&Ms development IMO is less awkward - a would-be proper class for B&M is naturally the empty set via the i-operator, whereas Suppes forces it to be with a somewhat awkward set builder definition.) An advantage of the i-operator is that it allows the unique x such that... to be expressed independently of set theory, a fact they exploit throughout their development of logic. -- Norm http://us.metamath.org/email.html (Reply to author at this URL. The from address in this post is not valid.) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) The i-operator ix phi, the (unique) x such that phi, is used in > Bell/Machover, _A Course in Mathematical Logic_ (1977), as a way to > introduce what they call virtual terms (p. 106) in logic. Later, in > set theory (p. 465), they define the set builder as {x : phi} =df iz(Ax(x e z <-> phi)) where z is not free in phi. So, for example, the union of two sets > could be equivalently defined as either A u B = iz(Ax(x e z <-> x e A v x e B)) or A u B = {x : x e A v x e B} > Right. (Though I don't know the exact approach of Bell/Machover. If you use Frege's approach there's nothing virtual about the term ix phi, even though it may not denote the phi [in case there is none, or there is more then one phi]. In those cases it denote the dummy element.) Anyway, in the context of ZFC we may define {x e A : phi} =df iz(Ax(x e z <-> x e A & phi)) , and in this case we actually have (for any A and any phi): z e {x e A : phi} <-> Ax(x e z <-> x e A & phi) from the axiom (schema) of separation and the axiom of extensionality [and the axiom schema(s) for the i-Operator]. For set theory definitions, they usually resort to the latter since it > is more compact, but occasionally they revert to the i-operator where it > is more expedient such as in the definition of function value. > Right. f(x) =df iy((x,y) e f). :-) Note that their set builder definition doesn't allow expressing proper > classes, since it would evaluate to the empty set in such a case. > Right. This follows from the i-Operator axiom (or theorem) ~E!xPhi[x] -> ixPhi[x] = 0 (at least in my approach). (In their entire ZFC development, they avoid speaking of proper classes by > using wff metavariables in place of the more typical class variables of > other authors. Suppes does this also, although B&Ms development IMO is > less awkward - a would-be proper class for B&M is naturally the empty > set via the i-operator, whereas Suppes forces it to be with a somewhat > awkward set builder definition.) > Right. (It's a pity that Suppes did not take up the i-Operator. In his book Introduction to Logic he just mentions that device VERY briefly in a footnote!) :-( An advantage of the i-operator is that it allows the [...] x such that... > to be expressed independently of set theory, a fact they exploit through- > out their development of logic. > Yes, of course. It's not a notion of set theory, but of logic. Hence my personal preference is an extension of standard FOPL_with_identity to comprise the i-Operator - especially when we set up that system for certain applications (say in arithmetic and/or set theory). In those systems there are natural choices for the dummy object: the number 0 in arithmetic and the empty set in set theory. B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> An advantage of the i-operator is that it allows the [...] x such that... >> to be expressed independently of set theory, a fact they exploit through- >> out their development of logic. > Yes, of course. It's not a notion of set theory, but of logic. Hence my > personal preference is an extension of standard FOPL_with_identity to > comprise the i-Operator - especially when we set up that system for > certain applications (say in arithmetic and/or set theory). In those > systems there are natural choices for the dummy object: the number 0 > in arithmetic and the empty set in set theory. > We might even go further and take the bold step to add Hilbert's e-Operator to this system. (I.e. extend the language with e ... and add the axiom schema ExPhi[x] -> Phi[exPhi[x]]. We may read an expression of the form exPhi[x] as a Phi. The axiom then just means: If there are Fs then an F is F. Sounds reasonable. Of course in the context of a decent set theory (i.e. ZF) this approach would imply the AC. (Hence curse and blessing at the same time?) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <6r9284dm7tkgtoglffadpqbarnbj6abqgm@4ax.com> <9ln98498dth8al10t6spiuci9sqo2qsdl2@4ax.com> posting-account=ZsXvqAoAAABmATPp--wbAhLaSprxKuin Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > ... something that has always put me off using an i-Operator. The funny thing is that many mathematicians use it freely in ordinary > (non formal) lingo of math. For example they formulate things like: æ æ æ æ ... when we define AuB to be the set of all x such that > æ æ æ æ æx is an element of A or x is an element of B we do need > æ æ æ æ æto show from the axioms that there is such a set. æ æ æ æ (David C. Ullrich in a recent post) Well, actually, using Frege's approach and a formal theory of the > i-Operator, we do not. ;-) Anyway, this definition can be formalized the following way: æ æ æ æ A u B =df iy(Ax(x e y <-> x e A v x e B)). Later we can show (from the axioms) that for any sets A and B there > actually is exactly one set of all x such that x is an element of A or x > is an element of B. In symbols: æ æ æ æ E!y(Ax(x e y <-> x e A v x e B)). And with this theorem (and the definition mentioned above) we get > immediately from the axioms of the i-operator: æ æ æ æ Ax(x e A u B <-> x e A v x e B) æfor any A, B. Imho the i-Operator would (to a certain extend) just reflect normal > mathematical practice (at a formal level). B. Am I being pedantic to point out that, at least under the standard ZF axioms, the definition of the union of two sets is done extremely differently than you have done it here? (I'm not familiary with the details of Frege's approach.) In normal ZF, of course, we call on the axioms of pairing and union, which are constructive axioms: they produce a new set. Pairing: forall A,B, exists S = {A,B} or: forall A (forall B (exists S (x in S iff (x = A or x = B)))) Union: forall S exists T (x in T iff exists X in S (x in X)) Then it is a theorem of ZF that for all sets A and B, there exists a set (unique by extentionality) whose elements are exactly those things which are elements of A, B, or both. You can't get there from logical considerations alone in ZF is my main point: just because x in A or x in B is a well-formed sentence (using only the primitives of set theory) for any A, B, and x does not mean that we automatically get a set of all objects x for which this sentence evaluates true. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Am I being pedantic to point out that, at least under the standard ZF > axioms, the definition of the union of two sets is done extremely > differently than you have done it here? > In no way! Though I don't get your point. :-) Note that besides the usual /axiom of union/ (U-Ax) sometimes the following axiom is adopted (u-Ax): EuAz(z e u <-> z e x v z e y). (See P. Suppes, Axiomatic Set Theory, for example.) E!uAz(z e u <-> z e x v z e y). Which (directly) justifies the _standard_ definition x u y = u <-> Az(z e u <-> z e x v z e y). But when adopting the method of definition mentioned in my former post, we don't need all this apparatus. :-) We just can formulate the definition: A u B =df iy(Ax(x e y <-> x e A v x e B)). That's the beauty of the i-Operator (in the way Frege introduced it)! Note: If there is exactly one x such that F(x) holds, den 'ixF(x)' just denotes that x, and if there is no x or more than one x such that F[x] holds, then 'ixF[x]' denotes the null set. (This way the term 'ixF(x)' always denotes a certain object/set.) You can't get there from logical considerations alone in ZF is my main > point: just because x in A or x in B is a well-formed sentence (using only the primitives of set theory) > for any A, B, and x does not mean that we automatically get a set of > all objects x for which this sentence evaluates true. > Right. Hint: read my former post again (especially the part AFTER stating the definition mentioned above). If you have questions you might ask. B. P.S. I'm not familiary with the details of Frege's approach. > I see. Hence your complaints. Just relax! The things I've explained are (logically and mathematically) sound. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <6r9284dm7tkgtoglffadpqbarnbj6abqgm@4ax.com> <9ln98498dth8al10t6spiuci9sqo2qsdl2@4ax.com> <974a84548vldegu6ub89nseq6afegede8h@4ax.com> posting-account=ZsXvqAoAAABmATPp--wbAhLaSprxKuin Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > Note that besides the usual /axiom of union/ (U-Ax) sometimes the > following axiom is adopted (u-Ax): æ æ æ æ EuAz(z e u <-> z e x v z e y). (See P. Suppes, Axiomatic Set Theory, for example.) > æ æ æ æ E!uAz(z e u <-> z e x v z e y). Which (directly) justifies the standard definition æ æ æ æ x u y = u <-> Az(z e u <-> z e x v z e y). But when adopting the method of definition mentioned in my former post, > we don't need all this apparatus. :-) We just can formulate the definition: æ æ æ æ A u B =df iy(Ax(x e y <-> x e A v x e B)). That's the beauty of the i-Operator (in the way Frege introduced it)! Note: If there is exactly one x such that F(x) holds, den 'ixF(x)' just > denotes that x, and if there is no x or more than one x such that F[x] > holds, then 'ixF[x]' denotes the null set. (This way the term 'ixF(x)' > always denotes a certain object/set.) > Oh, and one other thing: these axioms of union can't be the *usual* axioms of union, because they contain unbound variables. In fact, I think there's something seriously wrong with the first one u, which has the aftermentioned properties with respect to all sets z and unquantified sets x and y. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <6r9284dm7tkgtoglffadpqbarnbj6abqgm@4ax.com> <9ln98498dth8al10t6spiuci9sqo2qsdl2@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > We just can formulate the definition: A u B =df iy(Ax(x e y <-> x e A v x e B)). these axioms of union can't be the *usual* > axioms of union, because they contain unbound variables. I take it as implicit that there are, though tacit in display, universal quantifiers to close the formula. That's along the same lines as informally stating theorems and axioms as open formuals, such as x+0=x, when it is understood that the actual theorem or axiom is Ax x+0 = x. I think I've seen some authors actually state that caveat. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Oh, and one other thing: these axioms of union can't be the *usual* > axioms of union, because they contain unbound variables. > *sigh* We adopted the common practice to omit leading universal quantifiers. What was meant is: AxAyEuAz(z e u <-> z e x v z e y). Though the following (weaker) formulation would do too: (u-Ax) AxAyEuAz(z e x v z e y -> z e u). For any two sets x and y there's a set which contains [at least] all elements contained in x and y. This axiom replaces the axiom of pairs - if adopted. (For any two sets x and y we have from the power set axiom [and ...] the existence of {x} and {y}. And hence from the u-Ax [and ...] the existence of {x, y} := {x} u {y}.) In fact, I think there's something seriously wrong with the first one > u, which has the aftermentioned properties with respect to all sets z > and unquantified sets x and y. > Again, read: AxAyE!uAz(z e u <-> z e x v z e y). B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <6r9284dm7tkgtoglffadpqbarnbj6abqgm@4ax.com> <9ln98498dth8al10t6spiuci9sqo2qsdl2@4ax.com> <974a84548vldegu6ub89nseq6afegede8h@4ax.com> posting-account=ZsXvqAoAAABmATPp--wbAhLaSprxKuin Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > Am I being pedantic to point out that, at least under the standard ZF > axioms, the definition of the union of two sets is done extremely > differently than you have done it here? In no way! Though I don't get your point. :-) Note that besides the usual /axiom of union/ (U-Ax) sometimes the > following axiom is adopted (u-Ax): æ æ æ æ EuAz(z e u <-> z e x v z e y). (See P. Suppes, Axiomatic Set Theory, for example.) > æ æ æ æ E!uAz(z e u <-> z e x v z e y). Which (directly) justifies the standard definition æ æ æ æ x u y = u <-> Az(z e u <-> z e x v z e y). But when adopting the method of definition mentioned in my former post, > we don't need all this apparatus. :-) We just can formulate the definition: æ æ æ æ A u B =df iy(Ax(x e y <-> x e A v x e B)). That's the beauty of the i-Operator (in the way Frege introduced it)! Note: If there is exactly one x such that F(x) holds, den 'ixF(x)' just > denotes that x, and if there is no x or more than one x such that F[x] > holds, then 'ixF[x]' denotes the null set. (This way the term 'ixF(x)' > always denotes a certain object/set.) You can't get there from logical considerations alone in ZF is my main > point: just because æ æx in A or x in B is a well-formed sentence (using only the primitives of set theory) > for any A, B, and x does not mean that we automatically get a set of > all objects x for which this sentence evaluates true. Right. Hint: read my former post again (especially the part AFTER stating the > definition mentioned above). If you have questions you might ask. B. P.S. I'm not familiary with the details of Frege's approach. I see. Hence your complaints. Just relax! The things I've explained are > (logically and mathematically) sound. All right... so refresh my memory then... is there a natural-language approximation of how this i-operator should be read? (IIRC we started off on this tangent when we were talking about the various senses of to exist...and the i-operator is basically a third logical operator having something to do with existence but not covered by the E-operator?) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) All right... so refresh my memory then... is there a natural-language > approximation of how this i-operator should be read? > Right. That's the very point of that approach: it reflects the usage of _the_ in usual/normal language. Hence 0 =df iy(Ax(x !e y)) translates to: 0 is _the_ empty set. (if we read Ax(x !e y) as: y is an empty set.) etc. (Now one might ask: what the hell is _the_ present king of France? :-) IIRC we started off on this tangent when we were talking about the > various senses of to exist... > We did? and the i-operator is basically a third logical operator having something > to do with existence but not covered by the E-operator? > No. It's true, it's a variable binding operator (like A and E), but its result is not a formula (or a sentence) but a _term_. (See example above.) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > Am I being pedantic to point out that, at least under the standard ZF > axioms, the definition of the union of two sets is done extremely > differently than you have done it here? In no way! Though I don't get your point. :-) Note that besides the usual /axiom of union/ (U-Ax) sometimes the > following axiom is adopted (u-Ax): EuAz(z e u <-> z e x v z e y). That is not adopted as an axiom in ZF, which was the point being made. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> EuAz(z e u <-> z e x v z e y). > That is not adopted as an axiom in ZF, which was the point being made. > It is. (Hint: It replaces the axiom of pairs. The latter can be proved from the other axioms. Exercise!) LEARN TO READ: (See P. Suppes, Axiomatic Set Theory, for example.) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > EuAz(z e u <-> z e x v z e y). > That is not adopted as an axiom in ZF, which was the point being made. It is. (Hint: It replaces the axiom of pairs. The latter can be proved > from the other axioms. Exercise!) LEARN TO READ: (See P. Suppes, Axiomatic Set Theory, for example.) > I read quite well, thank you. And I do not read your version as any of the axioms stated at http://en.wikipedia.org/wiki/ZFC , the first 8 axioms/axiom_schemas of which form ZF. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) And I do not read your version as any of the axioms stated at > http://en.wikipedia.org/wiki/ZFC , > the first 8 axioms/axiom_schemas of which form ZF. > So what? There are several different axiom systems for ZF(C) out there. There's no such thing as a canonical set of axioms. (Though of course some axioms actually are part of the usual canon. :-) Maybe you already noticed: sometimes a null set axiom Null Set: Ex~Ey(y e x) is adopted. For example see: http://plato.stanford.edu/entries/set-theory/ZF.html Now this axiom is also not mentioned in the Wikipedia entry. Does that mean that _Jech_ does not know the axioms of ZF? :-) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > And I do not read your version as any of the axioms stated at > http://en.wikipedia.org/wiki/ZFC , > the first 8 axioms/axiom_schemas of which form ZF. So what? So that your version is not the only one. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > And I do not read your version as any of the axioms stated at > http://en.wikipedia.org/wiki/ZFC , > the first 8 axioms/axiom_schemas of which form ZF. So what? There are several different axiom systems for ZF(C) out there. > There's no such thing as a canonical set of axioms. (Though of course > some axioms actually are part of the usual canon. :-) Maybe you already noticed: sometimes a null set axiom Null Set: > Ex~Ay(y e x) is adopted. For example see: > http://plato.stanford.edu/entries/set-theory/ZF.html Now this axiom is also not mentioned in the Wikipedia entry. Does that > mean that Jech does not know the axioms of ZF? :-) It as you who seemed to be claiming that your axioms, or posssibly Suppes' axioms, were the cannonical ones. If there is no cannonical formulation, then no one is bound to follow your formulation, or Suppes', for the matter, === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > > It is you who seemed to be claiming that your axioms, or possibly > Suppes' axioms, were the canonical ones. > Huh?! ---------------------------------------------------------- Note that besides the usual /axiom of union/ (U-Ax) sometimes the following axiom is adopted (u-Ax): EuAz(z e u <-> z e x v z e y). (See P. Suppes, Axiomatic Set Theory, for example.) ---------------------------------------------------------- Does that sound as if I claimed that this axiom is standard? :-o ??? If there is no canonical formulation, then no one is bound to follow > your formulation, or Suppes', for the matter, > Right. Who the hell claimed otherwise? :-o B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) [...] There are several different axiom systems for ZF(C) out there. > There's no such thing as a canonical set of axioms. (Though of course > some axioms actually are part of the usual canon. :-) > The small union axiom: (u-Ax) AxAyEuAz(z e x v z e y -> z e u). For any two sets x and y there's a set which contains [at least] all elements contained in x and y. This axiom replaces the axiom of pairs - if adopted. The latter can be proved from the other axioms. *** Exercise! *** B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> >> Well-formed is determined by the rules for forming formulae, and my >> rules allow it. > Ok, that's fine. So now let's consider the formula Card(A) <= Card(B). Now please tell me if this formula uses your SINGLE predicate [Card(.) > <= Card(.)], or is a sentence formed by the two predicates [Card(.)] and > [. <= .]? The latter would allow to eliminate the terms using their > definitions, while in the former case this would be a logical error. (In > this case I would have to use the definition of [Card(.) <= Card(.)].) > Of course the solution would be to introduce what we would call /namespaces/ in programming. :-) With other words, in the context of a certain chapter it might mean [Card(A) <= Card(B)] , and in the context of the rest of the book, it might mean what we usually mean by it (speaking of cardinal numbers), namely that the cardinal number of A is smaller than or equal to the cardinal number of B. :-) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <9f6384553svvbc0ojddloc5e68djfrhkti@4ax.com> posting-account=ZsXvqAoAAABmATPp--wbAhLaSprxKuin Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > [...] That is the whole point: We're defining and entire string, > any part of which is meaningless on its own. Ok, got your point. And it does make sense, to a certain extend. On the > other hand I already qualified this to be a very dangerous/misleading > approach. (See posts by MoeBlee and me in this thread.) Go look up the discussion of this very idea in either Hrbacek/Jech... And they IMMEDIATELY got trapped by that misleading notation! After > stating some of the properties of |.| <= |.|, they write: We see that <= is reflexive and transitive. Of course we DON'T see that. After all <= is not even defined at that > stage! What we see is that |.| <= |.| is reflexive and transitive. See?! (That's exactly my point!) B. The mentioned problems can be completely avoided by simply defining æ æ æ æ x << y :<-> Ef f is an injection from x into y > and > æ æ æ æ æx ~ y :<-> Ef f is an injection from x onto y instead (and abandoning to mention card as a proper substring in a > predicate-symbol consisting of a string). discussion. And yes, I definitely understand your and MoeBlee's larger point. In some ways, this is an example of a technique which is more pedagogical than anything else -- that is, getting students to recall precisely what is meant by a given symbol or string or predicate, even if that predicate is written in a highly suggestive notation. (Though as you point out, Hrbacek/Jech themselves went beyond their own definitions because of how suggestive they were.) /As is probably obvious, I'm not a fully qualified mathematician, so take what I say with a grain of salt === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > And yes, I definitely understand your and MoeBlee's larger > point. granted that now I think I see how we could rigorously describe an elimination of Virgil's formula. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Perhaps inflexible would have been a better descriptor. Right. Rigor is a proper attribute to describe the logical approach. Tedious simple-minded pedantry also comes to mind. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Perhaps inflexible would have been a better descriptor. >> Right. Rigor is a proper attribute to describe the logical approach. Tedious simple-minded pedantry also comes to mind. Some comments related to the card subthread: 1. I don't see any theoretical problem with defining the compound expression Card(x) <= Card(y) as long as we recognize it as compound and eliminate it accordingly. There may be a danger of accidentally treating the component Card(x) on its own, which could be a possible practical drawback. (By the way, Metamath is flexible enough to use this definition if we wish, if we take appropriate care such as using a new unique symbol for <=, but I chose not to in its present development of set theory. While I won't argue that Metamath isn't Tedious simple-minded pedantry :) - indeed that is exactly what it is intended to be - this would not be the reason.) 2. Equinumerosity can be defined as either there exists a bijection or as A <= B & B <= A where A <= B means there exists an injection from A to B. Most textbooks use the former definition. Quine uses the latter: _Set Theory and Its Logic_ p. 78, but he discusses other possibilities (p. 208) and recognizes of course that the Schroeder-Bernstein theorem is needed to get the bijection version from his. 3. Metamath uses the smallest equinumerous ordinal definition of card, as was pointed out, but it also shows that the definition based on equinumerous sets of least rank is possible: http://us.metamath.org/mpegif/karden.html using Scott's trick: http://us.metamath.org/mpegif/scottex.html to show its existence: http://us.metamath.org/mpegif/kardex.html 4. Regarding the unique set such that... (iota) that was discussed expressed in ZFC, along with an extended related concept, Hilbert's epsilon. http://us.metamath.org/downloads/megillaward2005he.pdf -- Norm http://us.metamath.org/email.html (Reply to author at this URL. The from address in this post is not valid.) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) 1. I don't see any theoretical problem with defining the compound > expression Card(x) <= Card(y) as long as we recognize it as compound > and eliminate it accordingly. > But that's exactly my point! If we (later) introduce /Card/ and /<=/ in isolation we would get/have a problem, no? Though in the context of a single document (i.e. a book), a solution might be to introduce what we would call /namespaces/ in programming. With other words, in the context of a certain chapter Card(A) <= Card(B) might mean [Card(A) <= Card(B)] , i.e. the compound expression, and in the context of the rest of the document, it might mean what we usually mean by it, namely that the cardinal number of A is smaller than or equal to the cardinal number of B (i.e. a formula consisting of two terms and relation symbol). B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <4cj184h0gi8fvstcbs0q5531cgketbsd72@4ax.com> <6nt284t88n5dmjgi2keacsk04e7fsl75uc@4ax.com> <874p6k7k22.fsf@alatheia.dsl.inet.fi> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > Some comments related to the card subthread: I was hoping to exit this subthread, but I feel compelled to post, since I was one who first mentioned Metamath and Megill -- the creator of Metamath -- is here. > 1. I don't see any theoretical problem with defining the compound > expression Card(x) <= Card(y) as long as we recognize it as compound > and eliminate it accordingly. æThere may be a danger of accidentally > treating the component Card(x) on its own, which could be a possible > practical drawback. æ(By the way, Metamath is flexible enough to use > this definition if we wish, if we take appropriate care such as using a > new unique symbol for <=, but I chose not to in its present > development of set theory. æWhile I won't argue that Metamath isn't > Tedious simple-minded pedantry :) - indeed that is exactly what it is > intended to be - this would not be the reason.) Therefore, both Virgil's and MoeBlee's (via Suppes) definition can be valid, since they both are eliminable if necessary. Also, we see in this subthread, Aatu agrees with Virgil in that excessive rigor can be pedantic in certain situations. > 4. Regarding the unique set such that... æ(iota) that was discussed > expressed in ZFC, along with an extended related concept, Hilbert's > epsilon. æhttp://us.metamath.org/downloads/megillaward2005he.pdf Interesting. often refer to your site for proofs and definitions, as you can see. (I even mentioned Metamath in a thread regarding the definition of pi and e very recently!) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <4cj184h0gi8fvstcbs0q5531cgketbsd72@4ax.com> <6nt284t88n5dmjgi2keacsk04e7fsl75uc@4ax.com> <874p6k7k22.fsf@alatheia.dsl.inet.fi> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Also, we see in this subthread, Aatu agrees with > Virgil in that excessive rigor can be pedantic > in certain situations. And thus Aatu agrees with not just Virgil, but with what MoeBlee (and I should think, Balthasar) too have held in that regard, since the beginning of the discussion. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> Also, we see in this subthread, Aatu agrees with >> Virgil in that excessive rigor can be pedantic >> in certain situations. > And thus Aatu agrees with not just Virgil, but with what MoeBlee (and > I should think, Balthasar) too have held in that regard, since the > beginning of the discussion. > I second that. On the other hand, I just see a certain value in being pedantic from time to time; especially in the context of mathematical logic and a treatment of mathematics as a formal system (for example in the context of axiomatic set theory). And certainly /sci.logic/ has its place for such pedantry. :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > Perhaps inflexible would have been a better descriptor. Right. Rigor is a proper attribute to describe the logical approach. Tedious simple-minded pedantry also comes to mind. There is a place for such rigor, but the place is not in dealing with such as WM. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > In particular, WM states that there ought not to be more > reals than naturals. With Robinson's hyperreals, the set of > hypernaturals and the set of hyperreals have exactly the > same cardinality. What does that have to do with there being no more reals than naturals? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Since I am a mathematical fictionalist, I agree with WM that THERE ARE > NO (infinite) SETS _in reality_. So what? What would it mean for sets to exist _in reality_? That is, what is the difference between sets, numbers and so on, existing as parts of the mathematical stories we find compelling, and their ACTUALLY existing? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor On 20 Jul 2008 15:35:17 +0300, Aatu Koskensilta >> Since I am a mathematical fictionalist, I agree with WM that THERE ARE >> NO (infinite) SETS _in reality_. So what? > What would it mean for sets to exist _in reality_? That is, what is > the difference between sets, numbers and so on, existing as parts of > the mathematical stories we find compelling, and their ACTUALLY > existing? > Depends. IF we had some sort of awareness of those entities (if they existed) -G.9adel called it intuition, iirc- then this would mean that our stories (or at least some of them) are not just made up as we like them, but that they have been shaped/inspired by the TRUTH (i.e. the reality). :-) Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have any less confidence in this kind of perception, i.e. in mathematical intuition, than in any sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in the future. The set-theoretical paradoxes are hardly more troublesome for mathematics than deceptions of the senses are for physics ... New mathematical intuitions leading to a decision of such problems as Cantor's continuum hypothesis are perfectly possible. (Kurt G.9adel, What is Cantor.89s continuum problem?, 1964) By physical reality I mean the material world, ... the world which physical science tries to describe. ... For me, and I suppose for most mathematicians, there is another reality, which I will call 'mathematical reality'; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it, others that it is outside and independent of us. A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all. I should not wish to argue any of these questions here even if I were competent to do so, but I will state my own position dogmatically in order to avoid minor misapprehensions. I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems we prove, and which we describe grandiloquently as our creations are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards ... (G.H. Hardy, A Mathematician's Apology, 1940) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor <87zloc7myi.fsf@alatheia.dsl.inet.fi> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On 20 Jul., 14:35, Aatu Koskensilta the difference between sets, numbers and so on, existing as parts of > the mathematical stories we find compelling, and their ACTUALLY > existing? Cantor an Mittag-Leffler, 22.9.1884 ... Mit diesen Ideen einer genaueren Ergr.9fndung des Wesens alles Organischen besch.8aftige ich mich schon seit 14 Jahren, sie bilden die eigentliche Veranlassung, weshalb ich das m.9fhsame und wenig Dank verheissende Gesch.8aft der Untersuchung von Punctmengen unternommen und in diesem Zeitraum keinen Augenblick aus den Augen verloren habe. Ausserdem interessirt mich rein theoretisch das Wesen des Staates und was dazu geh.9art, weil ich auch dar.9fber meine Gesichtspuncte habe, die zu mathematischer Formulierung sp.8ater f.9fhren d.9frfte; das Auffallende, was Sie darin vielleicht finden, verschwindet, wenn Sie erw.8agen, dass auch der Staat ein organisches Wesen gewissermassen repr.8asentiert. The same goes for a hamburger (for instance). === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Cantor an Mittag-Leffler, 22.9.1884 > ... Mit diesen Ideen einer genaueren Ergr.9fndung des Wesens alles > Organischen besch.8aftige ich mich schon seit 14 Jahren, sie bilden die > eigentliche Veranlassung, weshalb ich das m.9fhsame und wenig Dank > verheissende Gesch.8aft der Untersuchung von Punctmengen unternommen und > in diesem Zeitraum keinen Augenblick aus den Augen verloren habe. > Ausserdem interessirt mich rein theoretisch das Wesen des Staates und > was dazu geh.9art, weil ich auch dar.9fber meine Gesichtspuncte habe, die > zu mathematischer Formulierung sp.8ater f.9fhren d.9frfte; das Auffallende, > was Sie darin vielleicht finden, verschwindet, wenn Sie erw.8agen, dass > auch der Staat ein organisches Wesen gewissermassen repr.8asentiert. The same goes for a hamburger (for instance). Alas, my German is not quite up to the task of making any sense of the quote you kindly provided. Perhaps you could explain, in English, what you take to be the difference between sets, numbers and so on, existing as parts of the mathematical stories we find compelling, and their ACTUALLY existing? In case of a hamburger, the difference is obvious: an actual hamburger we can eat. In case of sets, numbers and so on, the distinction is obscure since we do not think of sets, numbers and so on, as something we could eat, or interact with in any such manner. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor On 20 Jul 2008 17:16:24 +0300, Aatu Koskensilta [...] In case of sets, numbers and so on, the distinction is obscure > since we do not think of sets, numbers and so on, as something we > could eat, or interact with in any such manner. > Well, some eat with their eyes ... I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems we prove, and which we describe grandiloquently as our creations are simply our notes of our observations. (G.H. Hardy, A Mathematician's Apology, 1940) B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor <87zloc7myi.fsf@alatheia.dsl.inet.fi> <87iqv063pj.fsf@alatheia.dsl.inet.fi> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On 20 Jul., 16:16, Aatu Koskensilta ... Mit diesen Ideen einer genaueren Ergr.9fndung des Wesens alles > Organischen besch.8aftige ich mich schon seit 14 Jahren, sie bilden die > eigentliche Veranlassung, weshalb ich das m.9fhsame und wenig Dank > verheissende Gesch.8aft der Untersuchung von Punctmengen unternommen und > in diesem Zeitraum keinen Augenblick aus den Augen verloren habe. > Ausserdem interessirt mich rein theoretisch das Wesen des Staates und > was dazu geh.9art, weil ich auch dar.9fber meine Gesichtspuncte habe, die > zu mathematischer Formulierung sp.8ater f.9fhren d.9frfte; das Auffallende, > was Sie darin vielleicht finden, verschwindet, wenn Sie erw.8agen, dass > auch der Staat ein organisches Wesen gewissermassen repr.8asentiert. The same goes for a hamburger (for instance). Alas, my German is not quite up to the task of making any sense of the > quote you kindly provided. Sorry your permanent Wittgenstein quote made me believe that you understand German. I quoted from Cantor's letter to Mittag-Leffler. Cantor says that he had done all his research of (infinite) point sets mainly/really in order to discover the essense or suchness of all organic (existence). Thus he implied the possibility of a mapping of reality by set theory. It came to my mind that the hamburger that you mentioned recently also consists of organic substance. > Perhaps you could explain, in English, what > you take to be the difference between sets, numbers and so on, > existing as parts of the mathematical stories we find compelling, and > their ACTUALLY existing? First of all, I do not believe that there is a Platonic shelf where we can take numbers and theorems from. Both, numbers and theorems, are constructed by minds. After they have been constructed, they exist, at least in the mind that constructed them, or on the paper they are written on. They exist in the same way a hamburger exists. And in the same way their existence can cease. A construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection [13, p. 2]. When the objects of discussion are linguistic entities ... then that collection of entities may vary as a result of discussion about them. A consequence of this is that the 'natural numbers' of today are not the same as the 'natural numbers' of yesterday [14, p. 478]. And I may add, what differs is not constant, i.e, it is not a constant set. [13] Edward Nelson: Predicative Arithmetic, Princeton University Press (1986) 2. http://www.math.princeton.edu/~nelson/books/pa.pdf [14] David Isles: What evidence is there that 2^65536 is a natural number?, Notre Dame Journal of Formal Logic, Volume 33, Number 4, (1992) 465-480. http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093634... And from this form of existence it is clear that there are no alephs but at most oo. > In case of a hamburger, the difference is > obvious: an actual hamburger we can eat. In case of sets, numbers and > so on, the distinction is obscure since we do not think of sets, > numbers and so on, as something we could eat, or interact with in any > such manner. Like Dedekind, Cantor was doing mathematics in order to apply it to reality (in the above quote he even includes politics). He would never have accepted numbers that have no definition or cannot be named. Nor do I. And I think that mathematics including such things is in a bad shape. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080325 Ubuntu/7.10 (gutsy) Firefox/2.0.0.13,gzip(gfe),gzip(gfe) kappa+1 > kappa is true for every number kappa, regardless of > whether kappa is finite or infinite. That's not exactly an accurate characterisation, I think. What Tony means by the Axiomatic principle x+y>y <-> x>0 ... (1) is the observation that in school mathematics (i.e. arithmetic on the reals) this is true. Since school mathematics is something he did well at, and all he knows about, he naturally feels that anything with math in the title somewhere ought to behave *exactly* like school mathematics. I keep meaning to find out what he thinks about non- Abelian groups... ---------------- interlude ------------------- Tony! Consider the following multiplication table: (x * y = row x, col y) * | 1 2 3 4 5 6 ------------------ 1 | 1 2 3 4 5 6 2 | 2 3 1 6 4 5 3 | 3 1 2 5 6 4 4 | 4 5 6 1 2 3 5 | 5 6 4 3 1 2 6 | 6 4 5 2 3 1 Observe that this is a valid table for the axioms of a (multiplicative) group. So as well as multiply, you can do divide, because each produce appears exactly once in each row and column. Also observe that (e.g.) 4 * 5 = 2, whereas 5 * 4 =3, so it is not always true that x*y=y*x. Please say (one of): (a) I know what group this is. (b) I would like to know what this group is. (c) This is obviously schlock. (d) This doesn't look like an infinite set, so I'm not interested. ----------------------------------------------- > In ZFC, we know that this holds for finite cardinals, but it doesn't > hold for infinite cardinals -- because when we extend the definition > of addition to infinite cardinals, we find counterexamples to the > inequality kappa+1 > kappa. In other words, standard mathematicians > sacrifice the inequality kappa+1 > kappa in order to retain certain > other properties of + (especially that if card(x) + card(y) = card(z) > and x and y are disjoint, then there exists a bijection between > xuy and z). A simple answer to this would be to use some symbol other than '+'. Typically, cranks believe that each symbol they learnt in school has the meaning they learnt in school somehow inherent in it. Would this satisfy Tony? I don't think so, quite... > But TO doesn't want to sacrifice this. I think what TO really wants is a set theory with infinite sets to which the pigeonhole principle (PP) applies. > Therefore TO would > prefer a set theory other than ZFC in which to work. It may > be possible that there's a rigorous theory other than ZFC > which has most, if not all, of the properties that TO > desires in a set theory. He could certainly have any finitist set theory, and the PP would apply universally. But no, (I believe) he wants something with the word infinite in it (and I have pointed out that actually his notions fit extremely well with the idea of an *imponderably large, yet perfectly finite* entity, but he assures us he is confident his Big'un and friends are 'really infinite', whatever that means exactly). Could he have a set theory in which the PP works for infinite sets? Since the PP is more or less a defining characteristic of a finite set (in normal set theory), it's hard to see how. It is hard to understand the real point of these posts you make. *Of course* there not only could be but are lots of other areas of mathematics that have nothing to do with set theory. Of course, *anyone* is welcome to invent their own. But cranks do not want to show us something they have made (and accept criticism of aspects of it that they hadn't noticed), they just want to tell us that set theory is wrong. And overwhelmingly their arguments simply show ignorance of mathematics, ranging from slight to total (in the case of WM in particular). They reject notions like definition in favour of poetry, they sail past contradictions with a variety of sociomanundulatologisms, and so on. Just read any web Crank checklist to see how well they fit. Brian Chandler === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > For the minimal number A of symbols in at least one heap that is > required to distinguish a maximal number B of heaps in a set of heaps For every *finite* case, the maximum number, B, of heaps that can > be disinguished, is the number of symbols in at least one heap. Look! Over there! A Pink Elephant! > In every case - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) For the minimal number A of symbols in at least one heap that is > required to distinguish æa maximal number B of heaps in a set of heaps For every *finite* case, the maximum number, B, of heaps that can > be disinguished, is the number of symbols in at least one heap. For the minimal number A of symbols in at least one heap that is required to distinguish a maximal number B of heaps in a set of heaps we obtain following possibilities: 1) A = n in N <==> B = n in N. 2) A = n in N <==> B > n in N. 3) A > n in N <==> B = n in N. 4) A > n in N <==> B > n in N. Case (1) is obviously correct for finite heaps and sets. Cases (2) and (3) are impossible, because all natural numbers n are already absorbed by case (1). There is nothing left but case (4). === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > For the minimal number A of symbols in at least one heap that is > required to distinguish a maximal number B of heaps in a set of heaps Except that, among other things, what WM calls heaps are not heaps, but one-member multisets, each being like a function from some {1,2,3,...,n} to the singleton set {o}. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > For the minimal number A of symbols Look! Over There! A Pink Elephant! >in at least one heap - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) For the minimal number A of symbols Look! Over There! A Pink Elephant! in at least one heap The number of symbols in a set of heaps cannot surpass the number of symbols in one of the heaps (= elements of the set), because similar symbols cannot exist in a set other than being in one of its elements. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor For the minimal number A of symbols Look! Over There! A Pink Elephant! in at least one heap The number of symbols in a set of heaps cannot surpass the number of > symbols in one of the heaps (= elements of the set), because similar > symbols cannot exist in a set other than being in one of its elements. WM again relies only on his pseudo-theorem that what is true for all finite sets is true for all sets. But absent the assumption that all sets are finite, or something equivalent, WM cannot prove that all sets must be finite, nor prove that pseudotheorem. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > For the minimal number A of symbols Look! Over There! A Pink Elephant! in at least one heap The number of symbols in a set of heaps cannot surpass the number of > symbols in one of the heaps (= elements of the set), because similar > symbols cannot exist in a set other than being in one of its elements. Look! Over There! A Pink Elephant! This one of its elements has to be the same for every symbol. - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) > OK, I missed the definitions of chocolate, vanilla, and strawberry > sets, but it appears what Albrecht is trying to do is use what I > called the infinite case of finite induction -- if a formula is true > for every natural number, then it must necessarily hold for the > set of all natural numbers. This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. Albrecht has produced no such set theory, and I know of none. While there can be set theories in which there is no set of all natural > numbers, as far as I am aware, for any set theory which allows a set of > all naturals, the statement There is a natural larger than n is a > counter-example to Albrecht's if a formula is true for every natural > number, then it must necessarily hold for the set of all natural numbers. <= that's not from me. You are citing wrong!!! Albrecht S. Storz === Subject: Re: A consideration concerning the diagonal argument of G. Cantor OK, I missed the definitions of chocolate, vanilla, and strawberry > sets, but it appears what Albrecht is trying to do is use what I > called the infinite case of finite induction -- if a formula is true > for every natural number, then it must necessarily hold for the > set of all natural numbers. >> This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. >> Albrecht has produced no such set theory, and I know of none. >> While there can be set theories in which there is no set of all natural >> numbers, as far as I am aware, for any set theory which allows a set of >> all naturals, the statement There is a natural larger than n is a >> counter-example to Albrecht's if a formula is true for every natural >> number, then it must necessarily hold for the set of all natural numbers. <= that's not from me. You are citing wrong!!! Albrecht S. Storz ZF's universe is irregular (ZF's universe isn't universal else ZF would be complete, so another collection contains it as the Russell set). Skolemizing, it's countable (its model is countable), and well-ordered, irregular, contains itself, thus N E N. Whether the transfer principle applies for various predicates (i.e. if for each then for all at once, for every at once, for all together varies. For example collections of ordinals closed under succession and predecession over all limit ordinals are ordinals, but they're not in ZF (Burali-Forti). Sets of sets are sets, but they're not, in ZF. No set is defined by the predicate true in ZF, and that alludes to a transfer to vacuity, to where when something is true for all sets that it's true for the empty set, null, the void. That the set of natural integers contains an infinite integer (eg one-point compactification which is implicit anyways, a prime or composite at infinity, or in other perspectives half of them are infinite in the semi-infinite) is much the same realization as that the Russell set contains itself. As a sputnik of quantification, it doesn't take much to show that a wide variety of set theories have infinite natural integers. Ross F. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) > I don't think anyone thinks that a collection can include more > elements than it includes. > I don't know what you know about the thinking of others. But if you > don't think so as you indicate, you should be near to agree that there > could not be a number which is greater than any natural number and > depicts the quantity of the set of the natural numbers. > If there is a number which depicts the cardinality of the set of > naturals, it must be a number in the set. Since none of the naturals > is able to do this, there is no cardinality of N. > To claim, that a number which is greater than any of the naturals is > no way out since this number can not decribe the quantity of the > natural numbers in an adequate manner: If this number exists, it is > too great. > If you don't know what quantity is, I can't help. > Do you mean count, in the > sense that in any vanilla collection, since there is a left end and a > right end, we can label every element with a natural number starting > at 1, and the natural number labelling the element at the right end of > the vanilla collection is the count of the collection. Is this what > you mean by quantity? > Leave your vanillachichinella collections behind. OK, I missed the definitions of chocolate, vanilla, and strawberry > sets, but it appears what Albrecht is trying to do is use what I > called the infinite case of finite induction -- if a formula is true > for every natural number, then it must necessarily hold for the > set of all natural numbers. This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. The problem is: Set theory is the wrong approach. Mathematics has to start with (natural) numbers. And numbers lead to contrary consequences as e.g. ZFC-sets. E.g. there is no inifinte object possible (if an object is something which is completely determinated) . Albrecht S. Storz === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. > The problem is: Set theory is the wrong approach. I was thinking once again about the definitions of the neapolitan ice cream flavors of sets again. Based on what I read from this subthread -- but someone correct me if I'm wrong -- we have: A vanilla set is a finite set. A strawberry set is a countably infinite set. A chocolate set is an uncountable set. And so we see that in this subthread, Albrecht takes the properties of finite vanilla sets and uses induction to extend these properties to infinite sets, derives a contradiction, and therefore concludes: > E.g. there is no inifinte object possible (if an object is something > which is completely determinated) . And of course, then the standard mathematician tells him that there are no infinite vanilla sets, but there can be infinite strawberry sets. And N is such a set. > Mathematics has to start with (natural) numbers. Both this line of Albrecht's, as well as Archimedes Plutonium's lines regarding algebra vs. geometry, raise yet another issue regarding the great set theory debate. Why do standard mathematicians insist that set theory (or sometimes category theory) must be the foundation of all of our mathematics? Why can't one use PA (the arithmetic of natural numbers, as would Albrecht) or geometry (perhaps in the form of Hilbert's axioms, minus the Parallel Postulate since AP mentions all three geometries -- elliptic, hyperbolic, and Euclidean) as the foundation of math rather than set theory? (I assume that AP is including set theory as part of the algebra or Galois theory that he always decries.) So, at least with Albrecht and AP, the real debate is not between ZFC and a nonstandard set theory, but between set theory itself and a wholly different foundation for mathematics. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. > The problem is: Set theory is the wrong approach. I was thinking once again about the definitions of the > neapolitan ice cream flavors of sets again. Based on what I read from this subthread -- but someone > correct me if I'm wrong -- we have: A vanilla set is a finite set. > A strawberry set is a countably infinite set. > A chocolate set is an uncountable set. And so we see that in this subthread, Albrecht takes the > properties of finite vanilla sets and uses induction to > extend these properties to infinite sets, derives a > contradiction, and therefore concludes: E.g. there is no inifinte object possible (if an object is something > which is completely determinated) . And of course, then the standard mathematician tells him > that there are no infinite vanilla sets, but there can be > infinite strawberry sets. And N is such a set. Mathematics has to start with (natural) numbers. Both this line of Albrecht's, as well as Archimedes Plutonium's > lines regarding algebra vs. geometry, raise yet another issue > regarding the great set theory debate. Why do standard mathematicians insist that set theory (or > sometimes category theory) must be the foundation of all of > our mathematics? Not all standard mathematicians do! Where have you ever seen a claim that spoke for standard mathematicians as a whole that they insist that only set theory or category theory can be a foundation? > Why can't one use PA (the arithmetic of > natural numbers, as would Albrecht) or geometry (perhaps in > the form of Hilbert's axioms, minus the Parallel Postulate since > AP mentions all three geometries -- elliptic, hyperbolic, and > Euclidean) as the foundation of math rather than set theory? If you know a way to derive analysis from number theory or whatever, then by all means, go ahead! MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor So, at least with Albrecht and AP, the real debate is not > between ZFC and a nonstandard set theory, but between set > theory itself and a wholly different foundation for mathematics. Except that no one has been insisting that set theory is the foundation of mathematics, only that it is a part of mathematics in good standing. What WM and ALbrecht, and their ilk, have been claiming is that set theory is not any part of mathematics, and that set theory is internally inconsistent, neither of which claims can they justify. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > So, at least with Albrecht and AP, the real debate is not > between ZFC and a nonstandard set theory, but between set > theory itself and a wholly different foundation for mathematics. Except that no one has been insisting that set theory is the foundation > of mathematics, only that it is a part of mathematics in good standing. What WM and ALbrecht, and their ilk, have æbeen claiming is that set > theory is not any part of mathematics, and that set theory is internally > inconsistent, neither of which claims can they justify. If we agree that the number of distinguishable paths in the binary tree increases from level to level by just the number of nodes which are present in this level, then the number of paths at any height of the tree cannot be larger than the number of nodes above that position plus the number of initial paths (which is 1). This consideration fits exactly with Dedekind's infinity: Dedekind gives an intuitive argument for the existence of an infinite set. Starting with an object t of one's thoughts, a thought-thing, one successively gets new thought-things by the thought of t, the thought of the thought of t, ...; hence the set of all thought-things of a human being is infinite like the set of all levels of the binary tree and the set of all paths that run through different nodes at one of these levels., There is no other infinity. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor What WM and ALbrecht, and their ilk, have æbeen claiming is that set > theory is not any part of mathematics, and that set theory is internally > inconsistent, neither of which claims can they justify. If we agree that the number of distinguishable paths I, for one, do not agree that what WM misrepresents as distinguishability has any relevance whatsoever. In any complete infinite binary tree, in which every node has two child nodes, the cardinality of the set of paths through any node is equal to the the cardinality of the set of paths through any other node, and is equal to the the cardinality of the set of all functions from the set of all naturals to any set with exactly two members. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What WM and ALbrecht, and their ilk, have æbeen claiming is that set > theory is not any part of mathematics, and that set theory is internally > inconsistent, neither of which claims can they justify. If we agree that the number of distinguishable paths I, for one, do not agree that [....] has any relevance whatsoever. > That's the ultimate argument! > In any complete infinite binary tree, in which every node has two child > nodes, the cardinality æof the set of paths through any node is equal to > the the cardinality æof the set of paths through any other node, Of course, that's infinity. > and is > equal to the the cardinality æof the set of all functions from the set > of all naturals to any set with exactly two members. Another comfimation of firm belief. In German, some decade ago, we would have called it Nibelungentreue (unshakeable loyalty). WM === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > What WM and ALbrecht, and their ilk, have æbeen claiming is that set > theory is not any part of mathematics, and that set theory is internally > inconsistent, neither of which claims can they justify. If we agree that the number of distinguishable paths I, for one, do not agree that [....] has any relevance whatsoever. That's the ultimate argument! It is the only one that WM offers that is not fatally flawed. That WM sneers at it when others use it is therefore hypocritical. In any complete infinite binary tree, in which every node has two child > nodes, the cardinality æof the set of paths through any node is equal to > the the cardinality æof the set of paths through any other node, Of course, that's infinity. and is > equal to the the cardinality æof the set of all functions from the set > of all naturals to any set with exactly two members. Another comfimation of firm belief. In German, some decade ago, we > would have called it Nibelungentreue (unshakeable loyalty). In English, it is still called a theorem. The various stages of its proof, in at least ZFC and NBG, have all been presented here, and none of them have been refuted by WM or anyone else without imposing assumptions that do not hold in ZFC, or even ZF, and NBG. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080325 Ubuntu/7.10 (gutsy) Firefox/2.0.0.13,gzip(gfe),gzip(gfe) > This is false in ZFC, but that doesn't mean that there can't be a > set theory in which it's true. > The problem is: Set theory is the wrong approach. I was thinking once again about the definitions of the > neapolitan ice cream flavors of sets again. Based on what I read from this subthread -- but someone > correct me if I'm wrong -- we have: A vanilla set is a finite set. > A strawberry set is a countably infinite set. > A chocolate set is an uncountable set. Normal terminology doesn't work with cranks and idiots, because they don't understand it. I tried to use this set of terms, as a shorter way of writing: A collection that can be counted so the counting ends at a counting number A collection that you can start counting, and guarantee to reach *any specified element eventually, but which never ends A collection that isn't either of the first two (should we find that such exists Generally cranks reject such labels, because they believe that real maths uses sensible-looking words that they think they know the meaning of. Actually they prefer poetry, where the precise meaning of the words isn't as important as the aural flow. > And so we see that in this subthread, Albrecht takes the > properties of finite vanilla sets and uses induction to > extend these properties to infinite sets, derives a > contradiction, and therefore concludes: No. As soon as you let a crank let Albrecht use a word like finite you've lost. He hasn't a clue what it means in maths, and more importantly, couldn't specify exactly what it means to *him* either. If he could be persuaded to agree with my definition of a vanilla set, you might be able to have a conversation. Incidentally, the *whole* point of labels like vanilla is to avoid the i-word (and its concomitant f-word), so saying finite vanilla set is just silly. E.g. there is no inifinte object possible (if an object is something > which is completely determinated) . A better conclusion would be - of course - that he doesn't have a coherent notion of what induction is (another i-word, you see). Above, I said not cranks, but cranks and idiots. Here's a post from Mueckenheim that seems to sum it all up in three lines: ----------------------------- > Pigeon hole principle. > Which fails for infinite sets, as Hilbert's Hotel demonstrates WM: Wrong observation. There are as many pigeons as holes (guests as rooms). If the principle would fail for infinite sets, then bijections would be useless for infinite sets. ----------------------------- Brian Chandler === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > A vanilla set is a finite set. > A strawberry set is a countably infinite set. > A chocolate set is an uncountable set. > Normal terminology doesn't work with cranks and idiots, because they > don't understand it. I tried to use this set of terms, as a shorter > way of writing: > A collection that can be counted so the counting ends at a counting > number > A collection that you can start counting, and guarantee to reach *any > specified element eventually, but which never ends > A collection that isn't either of the first two (should we find that > such exists > Generally cranks reject such labels, because they believe that real > maths uses sensible-looking words that they think they know the > meaning of. Actually they prefer poetry, where the precise meaning of > the words isn't as important as the aural flow. with my definitions of the set flavors. > Above, I said not cranks, but cranks and idiots. Here's a post > from Mueckenheim that seems to sum it all up in three lines: > ----------------------------- > Pigeon hole principle. > Which fails for infinite sets, as Hilbert's Hotel demonstrates > WM: > Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. > ----------------------------- And this is similar to not just Albrecht's ideas, but to Tony Orlow's as well. You mentioned in the TO subthread that TO, WM (in the post that you quote above) and Albrecht all seem to desire some sort of infinite pigeonhole principle: > Could [TO] have a set theory in which the PP works for infinite > sets? Since the PP is more or less a defining characteristic of a > finite set (in normal set theory), it's hard to see how. I'm wondering whether someone might be able to make a slight change to the Axiom of Infinity or one of the other axioms of ZFC (possibly Powerset, Replacement Schema, etc.). One proves the existence of an infinite set, but this set isn't omega. (One may call it Big'Un per TO -- I prefer to call it alpha.) In this proposed theory (that I haven't formulated yet), one can't derive from the existence of this infinite set the existence of omega w/out a contradiction. Also, one can't form a bijection between itself and a proper subset of itself -- for any alleged bijection would fail to be a set in this theory (for the new axioms would prevent it from being a set -- think Illegal Set Formation in some nonstandard set theories.) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080325 Ubuntu/7.10 (gutsy) Firefox/2.0.0.13,gzip(gfe),gzip(gfe) from Mueckenheim that seems to sum it all up in three lines: > ----------------------------- > Pigeon hole principle. > Which fails for infinite sets, as Hilbert's Hotel demonstrates > WM: > Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. > ----------------------------- And this is similar to not just Albrecht's ideas, > but to Tony Orlow's as well. The only thing in common with these three individuals' ideas is that they are incoherent. I quoted Mueckenheim's post above because it seemed to be a distillation of incoherence into some sort of minimal piece of text. Try to work out what it means! Wrong observation - WM is not a native speaker of English, so perhaps he means not the obvious reading (You should have made a different observation, whatever that would mean), but rather False observation. In other words, despite the fact that there is a bijection from {0, 1, 2, ... } to {1, 2, 3, ...} somehow there isn't. Or there shouldn't be. Or anyway this gives an answer that WM doesn't like, so (skipping the direct jump to Set theory is inconsistent) bijections are useless (whatever that means). But is this similar to AS or TO's ideas? I suppose there is a sort of commonality in the view of things like PHP as facts about the universe. Anything that works for collections of beans or pieces of string just must be True. Yet this lack of understanding hardly leads to anything in common that you could call an idea - in particular, of course WM asserts that Infinite sets do not Exist, TO asserts that Infinite sets do Exist, and what's more, Infinite Naturals, the Full Monticopia. If there were any real ideas here, surely at least one of the cranks would be able to understand (and probably correct in detail) the ideas of at least one other crank. But it never happens. [I'll use PHP for Pigeonhole principle] You mentioned in the TO subthread that TO, WM > (in the post that you quote above) and Albrecht > all seem to desire some sort of infinite > pigeonhole principle: Could [TO] have a set theory in which the PP works for infinite > sets? Since the PP is more or less a defining characteristic of a > finite set (in normal set theory), it's hard to see how. I'm wondering whether someone might be able to make a > slight change to the Axiom of Infinity or one of the > other axioms of ZFC (possibly Powerset, Replacement > Schema, etc.). One proves the existence of an infinite > set, but this set isn't omega. (One may call it > Big'Un per TO -- I prefer to call it alpha.) In this > proposed theory (that I haven't formulated yet), one > can't derive from the existence of this infinite set > the existence of omega w/out a contradiction. Also, > one can't form a bijection between itself and a proper > subset of itself -- for any alleged bijection would > fail to be a set in this theory (for the new axioms > would prevent it from being a set -- think Illegal > Set Formation in some nonstandard set theories.) Omega, remember, is the set of (all) naturals. Of course you can have a theory without the set of all naturals, but it's extremely hard to see how it can include anything known as an infinite set, unless infinite has changed its meaning beyond all recognition. So explain what you mean by saying that alpha is an infinite set. Say it in different words, not starting with i- or f-. Again, the PHP boils down to saying - if two collections of different sizes are laid side by side, then if both ends match up, there must be a kink in the middle. Having two ends when laid out in a (discrete) line is a diagnostic for susceptibility to the PHP. And a normal infinite set precisely means one which (if it can be laid down in a discrete sequence) does not have two ends. You see, TO quite specifically argues all over the place about how this and that happens at the (second) end of the Tonats. To him, declaring the end of something endless is perfectly ok. (Of course he prefers to use the Latin for endless, hoping that others are as easily misled by it as he is.) Why not champion the possibility of triangles with four or more corners? Perhaps in a different geometry they could all be made to have internal angles that sum to 180 degrees... I expect you say absurd, but this is no different. After all, we could loudly assert that all four corners of a triangle have a distinct tripleness. Or perhaps each corner could have not a whole point, but just 3/4 of a point. (Recall that part of TO's attempt to rescue his beloved Principle of infinite induction, faced with a loooong, careful treatment of limits and the staircase problem involved points in TO's scheme having directions, and when that didn't quite work, multiple directions, always just happening to be in the direction Tony thought most useful for the current bit of argument in hand. No less ridiculous than 3/4-points, I submit.) Brian Chandler === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > The problem is: Set theory is the wrong approach. Mathematics has to > start with (natural) numbers. Geometry does a great deal without any natural numbers at all. And it wasn't until Peano that arithmetic could be reasonably certain about what natural numbers were. > And numbers lead to contrary > consequences as e.g. ZFC-sets. That they lead to results Albrecht does not like is not at all the same as leading to internal contradictions, which no one has been able to show for such set theories as ZF, ZFC or NBG. Albrecht is quite free to try to develop his own set theory based on his own axioms, but until he can show internal inconsistency in other set theories, which he has not even come close to doing, he should shut his yap about them. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > The problem is: Set theory is the wrong approach. Mathematics has to > start with (natural) numbers. Geometry does a great deal without any natural numbers at all. Okay. So I had said only the half thruth. And it wasn't until Peano that arithmetic could be reasonably certain > about what natural numbers were. Nonsense. And numbers lead to contrary > consequences as e.g. ZFC-sets. That they lead to results Albrecht does not like is not at all the same > as leading to internal contradictions, which no one has been able to > show for such set theories as ZF, ZFC or NBG. If the results, which are deduced from the properties of numbers, contradicts the results which are deduced from the properties of sets, this should be strongly considered. Albrecht is quite free to try to develop his own set theory based on his > own axioms, but until he can show internal inconsistency in other set > theories, which he has not even come close to doing, he should shut his > yap about them. I'm interested in the theory of discret quantities, not in set theory. Set theory is a wrong approach to base the mathematics. Albrecht S. Storz === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > The problem is: Set theory is the wrong approach. Mathematics has to > start with (natural) numbers. Geometry does a great deal without any natural numbers at all. > Okay. So I had said only the half thruth. > And it wasn't until Peano that arithmetic could be reasonably certain > about what natural numbers were. Nonsense. > And numbers lead to contrary > consequences as e.g. ZFC-sets. That they lead to results Albrecht does not like is not at all the same > as leading to internal contradictions, which no one has been able to > show for such set theories as ZF, ZFC or NBG. If the results, which are deduced from the properties of numbers, > contradicts the results which are deduced from the properties of sets, > this should be strongly considered. When Albrecht has given a comprehensive definitions of numbers and their properties, e.g., an axiom system for them which is sufficient for, say, analysis, but without any use of sets, only then will he have any right to make that claim. At present he has no such right. > Albrecht is quite free to try to develop his own set theory based on his > own axioms, but until he can show internal inconsistency in other set > theories, which he has not even come close to doing, he should shut his > yap about them. I'm interested in the theory of discret quantities, not in set theory. > Set theory is a wrong approach to base the mathematics. That is Albrecht's opinion. Set theory has been shown to be a satisfactory base at least for analysis and a large part of abstract algebra. Whether it is satisfactory as a basis for the whole of mathematics is still an open question, at least among mathematicians. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > And it wasn't until Peano that arithmetic could be reasonably certain > about what natural numbers were. What sort of uncertainty do you think there was as to what the natural numbers are before Peano arithmetic? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor And it wasn't until Peano that arithmetic could be reasonably certain > about what natural numbers were. What sort of uncertainty do you think there was as to what the natural > numbers are before Peano arithmetic? The Peano postulates are the first brief but still coherent exposition of the minimal necessary properties of the naturals. At least the first one I am aware of. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > The Peano postulates are the first brief but still coherent exposition > of the minimal necessary properties of the naturals. At least the first > one I am aware of. Let's set aside historical questions about who should get the credit for the mathematical characteristion of the naturals. What sort of uncertainty as to what natural numbers are do you think there was before such a characterisation? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor The Peano postulates are the first brief but still coherent exposition > of the minimal necessary properties of the naturals. At least the first > one I am aware of. Let's set aside historical questions about who should get the credit > for the mathematical characteristion of the naturals. What sort of > uncertainty as to what natural numbers are do you think there was > before such a characterisation? Obviously there was some uncertainly about how to characterize them, which Peano's characterization largely settled. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) > I don't want to discuss the antinomy of infinity away. It is you who > thinks that there is no antinomy. We are quite aware than infinite sets often behave in counterintuitive > ways, but those intuitions are not only based on the finite naturals, > they are based on fairly small naturals. Humans, for example are supposedly not able to perceive directly the > number of any number of objects greater than five. However, careful logical analysis does not detect any > self-contradictions within several of the axiomatic set theories which > allow infinite sets. It's just because it is elaborated in this manner over a time of > nearly 100 years long. So there are seemingly no self-contradictions > in e.g. ZF. But this system don't fit to the properties of the natural > numbers as easy is to show: there is no quantity which depicts the > quantity of the natural numbers as the set theoretic cardinal number > of infinite sets suggests. Albrecht misses the point that what he wants to call 'the natural > numbers' an what ZF, NBG, etc., define as 'the natural numbers' need not > be the same thing. In mathematics, including set theories, formal definitions override > personal opinion. So that Albrecht's person opinions about what 'natural > numbers' should be like are irrelevant within, e.g., ZF or NBG. Maybe. But than I think, mathematics is on the total wrong way. Mathematics is based well only and only on the natural numbers as they are: X XX XXX ... If mathematics lost this base, mathematics is lost. Which fact encourages mathematicians to investigate such systems. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > I don't want to discuss the antinomy of infinity away. It is you who > thinks that there is no antinomy. We are quite aware than infinite sets often behave in counterintuitive > ways, but those intuitions are not only based on the finite naturals, > they are based on fairly small naturals. Humans, for example are supposedly not able to perceive directly the > number of any number of objects greater than five. However, careful logical analysis does not detect any > self-contradictions within several of the axiomatic set theories which > allow infinite sets. It's just because it is elaborated in this manner over a time of > nearly 100 years long. So there are seemingly no self-contradictions > in e.g. ZF. But this system don't fit to the properties of the natural > numbers as easy is to show: there is no quantity which depicts the > quantity of the natural numbers as the set theoretic cardinal number > of infinite sets suggests. Albrecht misses the point that what he wants to call 'the natural > numbers' an what ZF, NBG, etc., define as 'the natural numbers' need not > be the same thing. In mathematics, including set theories, formal definitions override > personal opinion. So that Albrecht's person opinions about what 'natural > numbers' should be like are irrelevant within, e.g., ZF or NBG. > Maybe. But than I think, mathematics is on the total wrong way. Mathemtics is based on agreement among mathematicians to be convinced by suitably rigorous arguments. Yours don't qualify as being anywhere near to suitably rigorous. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) I don't want to discuss the antinomy of infinity away. It is you who > thinks that there is no antinomy. We are quite aware than infinite sets often behave in counterintuitive > ways, but those intuitions are not only based on the finite naturals, > they are based on fairly small naturals. Humans, for example are supposedly not able to perceive directly the > number of any number of objects greater than five. However, careful logical analysis does not detect any > self-contradictions within several of the axiomatic set theories which > allow infinite sets. It's just because it is elaborated in this manner over a time of > nearly 100 years long. So there are seemingly no self-contradictions > in e.g. ZF. But this system don't fit to the properties of the natural > numbers as easy is to show: there is no quantity which depicts the > quantity of the natural numbers as the set theoretic cardinal number > of infinite sets suggests. Albrecht misses the point that what he wants to call 'the natural > numbers' an what ZF, NBG, etc., define as 'the natural numbers' need not > be the same thing. In mathematics, including set theories, formal definitions override > personal opinion. So that Albrecht's person opinions about what 'natural > numbers' should be like are irrelevant within, e.g., ZF or NBG. > Maybe. But than I think, mathematics is on the total wrong way. Mathemtics is based on agreement among mathematicians to be convinced by > suitably rigorous arguments. I don't think so. Mathematics is not dedicated to mathematicians, what ever that may be. Mathematics is a knowledge-pool of all peoples of the world. An there are no priests which have the right to say what is part of mathematics, or what is important in mathematics, or what is right or wrong in mathematics. The only authority in mathematics is the truth. And there is a truth. It is misleaded to think that only the absence of contradiction is a valuable guide. Albrecht S. Storz Yours don't qualify as being anywhere near to suitably rigorous. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor I don't want to discuss the antinomy of infinity away. It is you > who > thinks that there is no antinomy. We are quite aware than infinite sets often behave in > counterintuitive > ways, but those intuitions are not only based on the finite > naturals, > they are based on fairly small naturals. Humans, for example are supposedly not able to perceive directly > the > number of any number of objects greater than five. However, careful logical analysis does not detect any > self-contradictions within several of the axiomatic set theories > which > allow infinite sets. It's just because it is elaborated in this manner over a time of > nearly 100 years long. So there are seemingly no self-contradictions > in e.g. ZF. But this system don't fit to the properties of the > natural > numbers as easy is to show: there is no quantity which depicts the > quantity of the natural numbers as the set theoretic cardinal number > of infinite sets suggests. Albrecht misses the point that what he wants to call 'the natural > numbers' an what ZF, NBG, etc., define as 'the natural numbers' need > not > be the same thing. In mathematics, including set theories, formal definitions override > personal opinion. So that Albrecht's person opinions about what > 'natural > numbers' should be like are irrelevant within, e.g., ZF or NBG. > Maybe. But than I think, mathematics is on the total wrong way. Mathemtics is based on agreement among mathematicians to be convinced by > suitably rigorous arguments. I don't think so. Mathematics is not dedicated to mathematicians, what > ever that may be. Mathematics is a knowledge-pool of all peoples of > the world. An there are no priests which have the right to say what is > part of mathematics, or what is important in mathematics, or what is > right or wrong in mathematics. Then why does Albrecht arrogate to himself powers that he denies to any other individual or any group? > It is misleaded to think that only the absence of contradiction is a > valuable guide. Since the presence of a contradiction is surely a guide, how can its absence not be one? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > I don't think so. Mathematics is not dedicated to mathematicians, what > ever that may be. Mathematics is a knowledge-pool of all peoples of > the world. An there are no priests which have the right to say what is > part of mathematics, or what is important in mathematics, or what is > right or wrong in mathematics. Precisely. And because there are no priests, you are free to postulate your own axiom systems and start deriving theorems from them. What saves this from detriorating into total anarchy is the utility of the system. If you can derive *meaningful* theorems, then people will probably be persuaded more easily to subscribe to your theories. Completeness of the real numbers is a pretty important property to have, and without Dedekind and Cantor's work, you would lose that. > The only authority in mathematics is the truth. And there is a truth. This, I think, is highly debatable. What is the truth regarding the parallel postulate, for instance? Depending on which version you use, you end up with three distinct versions of geometry, Euclidian, Riemannian and Lobachevski-Bolyai. All three of them are useful, and all three of them are true. > It is misleaded to think that only the absence of contradiction is a > valuable guide. And that was not asserted by anyone! Absence of contradiction is certainly a *necessary* condition, but it is not sufficient. What was said was that you are free to develop your own theory any time you bloody well wish. So go for it. However, the muddled exposition thrown together by you and Herr Professor Muckenheim and others are not even free from contradiction. Far from it, in fact. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) I don't want to discuss the antinomy of infinity away. It is you who > thinks that there is no antinomy. We are quite aware than infinite sets often behave in counterintuitive > ways, but those intuitions are not only based on the finite naturals, > they are based on fairly small naturals. Humans, for example are supposedly not able to perceive directly the > number of any number of objects greater than five. However, careful logical analysis does not detect any > self-contradictions within several æof the axiomatic set theories which > allow infinite sets. It's just because it is elaborated in this manner over a time of > nearly 100 years long. So there are seemingly no self-contradictions > in e.g. ZF. But this system don't fit to the properties of the natural > numbers as easy is to show: there is no quantity which depicts the > quantity of the natural numbers as the set theoretic cardinal number > of infinite sets suggests. Albrecht misses the point that what he wants to call æ'the natural > numbers' an what ZF, NBG, etc., define as 'the natural numbers' need not > be the same thing. In mathematics, including set theories, formal definitions override > personal opinion. So that Albrecht's person opinions about what 'natural > numbers' should be like are irrelevant within, e.g., æZF or NBG. Maybe. But than I think, mathematics is on the total wrong way. Mathem[a]tics is based on agreement among mathematicians to be convinced by > suitably [stupid] arguments. I corrected two minor errors. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Math[e]matics is based on agreement among mathematicians to be convinced by > suitably [stupid] arguments. I corrected two minor errors. Wm corrected a typo ('mathmatics' to mathematics'), but then made a major error of his own by not realizing that it is his own stupidity, not that of mathematics or logic, which prevents him from being convinced by the logical arguments of mathematics. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) > A proof which is based on false assumptions. Would you please tell us your true assumptions and your system of > logic by which you prove things from those true assumptions, in a > way that we can algorithmically check whether a given statement is one > of the true assumptions and whether a given argument is allowed by > your system of logic? E.g.: > True assumptions: > - something exists > - put together one thing and another one thing gives two things (under > some special conditions) That's not an answer to my question: Maybe. But that is enough to find out that there is no number which quantify the naturals: X XX XXX ... If a gapless collection of natural numbers, starting with the number 1, has a number which quantifies this collection, this number is in the collection, else there is no such number. Any number which isn't in the collection in view is either too great or is unable to depict the quantity of the collection. Albrecht S. Storz > I said, in a way that we can > algorithmically check whether a given statement is one of the true > assumptions and whether a given argument is allowed by your system of > logic. I was speaking of mathematical assumptions; ones that can be > put in some mathematical language. And that is the entire set of assumptions from which you derive > calculus for the sciences? No, the explanation is quite simple IF you pay attention to the EXACT > definitions of the terms. You have only a horribly sloppy regard for > the exact definitions, so, of course, you can't understand why the > Lowenheim-Skolem result is not a contradiction. Oh yes, I know. It is not a contradiction. And if you squeeze your > eyes shut, all is black. Of course, you have no mathematical reply. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > If a gapless collection of natural numbers, starting with the number > 1, has a number which quantifies this collection, this number is in > the collection, else there is no such number. Still false Any number which isn't in the collection in view is > either too great Nope. - William Hughes === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > A proof which is based on false assumptions. Would you please tell us your true assumptions and your system of > logic by which you prove things from those true assumptions, in a > way that we can algorithmically check whether a given statement is one > of the true assumptions and whether a given argument is allowed by > your system of logic? E.g.: > True assumptions: > - something exists > - put together one thing and another one thing gives two things (under > some special conditions) That's not an answer to my question: Maybe. But that is enough to find out that there is no number which > quantify the naturals: So you say. Meanwhile, you have no system of yours that answers to my question. > X > XX > XXX > ... If a gapless collection of natural numbers, starting with the number > 1, has a number which quantifies this collection, this number is in > the collection, else there is no such number. Any number which isn't in the collection in view is either too great > or is unable to depict the quantity of the collection. You seem to be POSITING that if a number is not a member of a set then that number cannot be the cardinality of that set. Is that right? Anyway, you still would be better served to have axioms that can be formalized and a specificed system of formal logic. > I said, in a way that we can > algorithmically check whether a given statement is one of the true > assumptions and whether a given argument is allowed by your system of > logic. I was speaking of mathematical assumptions; ones that can be > put in some mathematical language. And that is the entire set of assumptions from which you derive > calculus for the sciences? No, the explanation is quite simple IF you pay attention to the EXACT > definitions of the terms. You have only a horribly sloppy regard for > the exact definitions, so, of course, you can't understand why the > Lowenheim-Skolem result is not a contradiction. Oh yes, I know. It is not a contradiction. And if you squeeze your > eyes shut, all is black. Of course, you have no mathematical reply. No reply by you. So, it still stands that your misunderstanding of Lowenheim-Skolem stems from your lack of understand the actual mathematics involved in it. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) A proof which is based on false assumptions. Would you please tell us your true assumptions and your system of > logic by which you prove things from those true assumptions, in a > way that we can algorithmically check whether a given statement is one > of the true assumptions and whether a given argument is allowed by > your system of logic? E.g.: > True assumptions: > - something exists > - put together one thing and another one thing gives two things (under > some special conditions) That's not an answer to my question: Maybe. But that is enough to find out that there is no number which > quantify the naturals: So you say. Meanwhile, you have no system of yours that answers to my > question. X > XX > XXX > ... If a gapless collection of natural numbers, starting with the number > 1, has a number which quantifies this collection, this number is in > the collection, else there is no such number. Any number which isn't in the collection in view is either too great > or is unable to depict the quantity of the collection. You seem to be POSITING that if a number is not a member of a set then > that number cannot be the cardinality of that set. Is that right? I SHOW that under some circumstances if a number is not member of a collection, that number cannot be the quantity of that collection. Anyway, you still would be better served to have axioms that can be > formalized and a specificed system of formal logic. I said, in a way that we can > algorithmically check whether a given statement is one of the true > assumptions and whether a given argument is allowed by your system of > logic. I was speaking of mathematical assumptions; ones that can be > put in some mathematical language. And that is the entire set of assumptions from which you derive > calculus for the sciences? No, the explanation is quite simple IF you pay attention to the EXACT > definitions of the terms. You have only a horribly sloppy regard for > the exact definitions, so, of course, you can't understand why the > Lowenheim-Skolem result is not a contradiction. Oh yes, I know. It is not a contradiction. And if you squeeze your > eyes shut, all is black. Of course, you have no mathematical reply. No reply by you. So, it still stands that your misunderstanding of Lowenheim-Skolem > stems from your lack of understand the actual mathematics involved in > it. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > You seem to be POSITING that if a number is not a member of a set then > that number cannot be the cardinality of that set. Is that right? I SHOW that under some circumstances if a number is not member of a > collection, that number cannot be the quantity of that collection. At least for finite positive integer natural numbers this is not difficult to see. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor You seem to be POSITING that if a number is not a member of a set then > that number cannot be the cardinality of that set. Is that right? I SHOW that under some circumstances if a number is not member of a > collection, that number cannot be the quantity of that collection. At least for finite positive integer natural numbers this is not > difficult to see. That it fails for the set of all naturals and its supersets is also not difficult to see, and in ZF some such sets are required to exist. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) A proof which is based on false assumptions. Would you please tell us your true assumptions and your system of > logic by which you prove things from those true assumptions, in a > way that we can algorithmically check whether a given statement is one > of the true assumptions and whether a given argument is allowed by > your system of logic? E.g.: > True assumptions: > - something exists > - put together one thing and another one thing gives two things (under > some special conditions) That's not an answer to my question: Maybe. But that is enough to find out that there is no number which > quantify the naturals: So you say. Meanwhile, you have no system of yours that answers to my > question. X > XX > XXX > ... If a gapless collection of natural numbers, starting with the number > 1, has a number which quantifies this collection, this number is in > the collection, else there is no such number. Any number which isn't in the collection in view is either too great > or is unable to depict the quantity of the collection. You seem to be POSITING that if a number is not a member of a set then > that number cannot be the cardinality of that set. Is that right? I SHOW that under some circumstances if a number is not member of a > collection, that number cannot be the quantity of that collection. Sure, it's the case as you say [emphasis added], UNDER SOME CIRCUMSTANCES. (Though you yourself have not shown anything in any reasonable mathematical sense of 'show'.) Meanwhile, there is no suggestion that it is the case that if a number is not a member of a set then that number cannot be the cardinality of that set. And THAT is what you'd have to show to refute - by the kind of argument you're trying to use - that there may not be infinite cardinalities. > Anyway, you still would be better served to have axioms that can be > formalized and a specificed system of formal logic. I said, in a way that we can > algorithmically check whether a given statement is one of the true > assumptions and whether a given argument is allowed by your system of > logic. I was speaking of mathematical assumptions; ones that can be > put in some mathematical language. And that is the entire set of assumptions from which you derive > calculus for the sciences? No, the explanation is quite simple IF you pay attention to the EXACT > definitions of the terms. You have only a horribly sloppy regard for > the exact definitions, so, of course, you can't understand why the > Lowenheim-Skolem result is not a contradiction. Oh yes, I know. It is not a contradiction. And if you squeeze your > eyes shut, all is black. Of course, you have no mathematical reply. No reply by you. So, it still stands that your misunderstanding of Lowenheim-Skolem > stems from your lack of understand the actual mathematics involved in > it. Again, no reply, as it still stands that you're ignorant of the actual details of Lowenheim-Skolem and thus your basic misunderstanding of it. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > You seem to be POSITING that if a number is not a member of a set then > that number cannot be the cardinality of that set. Is that right? I SHOW that under some circumstances if a number is not member of a > collection, that number cannot be the quantity of that collection. For every initial segment of Von Neumann naturals, including the whole set of them, the number of members is not a member. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Good authors take it for granted. The reasom is simple. How do we > obtain theorems of logic? By examples and by counter examples. This really is the nut of most of these arguments. In fact, that is / > not/ how mathematicians claim to obtain theorems of logic. It is not of interest what mathematicians claim. Similarly it is not of interest what farmers might claim about > farming, or musicians about playing music, or physicists about > physics, or engineers about engineering, or mothers about mothering, > etc. There is apparently no reason to listen to anyone besides Our > Royal Self, because We Know Better. The interest in claims of mathematicians depends on the quality of mathematics done by those mathematicians. When, once upon a time, Mathematica was a Queen, with no ambiguities and no arbitrariness in its foundations and its results, the opinions of those who admired and mastered that queen were considered valuable and were highly appreciated. Since Mathematica has become a whore, receptive to every arbitrary axiom and every form of nonsense like undefinable definitions, finished infinities and so on, Mathematica's results and the opinions of its adherents are of the same value, namely equal to that of the youngest natural number. Why > does A ==> B not imply -A ==> -B? Why? The simplest answer is: because it /isn't/ one of the rules of > formal logic that A ==> B implies ~A ==> ~B. What more need be said? First of all, *why* this is a rule of formal logic, of course. But it is not a rule of formal logic that A->B implies ~A->~B is not > a rule of formal logic: that is instead an observation one makes / > about/ the rules of logic. So I will take your question to be why > isn't A->B implies ~A->~B a rule of formal logic? Well, for one thing, given the /other/ rules of formal logic, if we > also include (A->B) implies (~A->~B), then A->B implies A <-> B; and > so the symbol -> would be redundant in our calculations. Also, it > would then follow from (A and B)->A that (A and ~A)<->A for all > statements A; i.e. all statements and their negations would be > provably false if we included that rule, independent of whatever > axioms one might choose. So that is one reason why A->B implies ~A->~B > is not a rule of formal logic. That explanation is reasonable, but an independent explanation would have been to say that A ==> B implies ~A ==> ~B is wrong because it does not fit reality. There are counter examples in reality. Because we have counter examples. > But for infinite numbers of objects or predicates these simpel rules > may change. I gave you an example. That is like saying (A or B) doesn't imply B; but sometimes this > simple rule may change. For example when we also have ~A, then ~A and > (A or B) implies B. But so what? It is still not the case that in formal logic /as a > general principle/, (A or B) implies B. Why? Simply: because it is not > a rule of formal logic that as a general principle, (A or B) implies > B. I did not ask what rules are applied but why they are applied. They are applied because they provide a framework wherein all parties > can agree regarding whether a given statement is true, given a fixed > set of other statements which have previously been agreed upon as > being true, by performing an unambiguous calculation. The results could be agreeable to all parties too, even if there were a different framework. In my opinion the reason for choosing the existing rules is their accordance with reality. So if we agree that all men are mortal, and we agree that Socrates is > a man, then it is simply a calculation which we perform to conclude > that we then agree: Socrates is mortal. And that corresponds to the reasoning process of most people. But you > generally do /not/ apply this sort of reasoning in your arguments; I apply just this kind of reasoning. For instance: If we agree that the number of distinguishable paths in the binary tree increases from level to level by just the number of nodes which are present in this level, then the number of paths at any height of the tree cannot be larger than the number of nodes above that position plus the number of initial paths (which is 1). === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Good authors take it for granted. The reasom is simple. How do we > obtain theorems of logic? By examples and by counter examples. This really is the nut of most of these arguments. In fact, that is / > not/ how mathematicians claim to obtain theorems of logic. It is not of interest what mathematicians claim. Similarly it is not of interest what farmers might claim about > farming, or musicians about playing music, or physicists about > physics, or engineers about engineering, or mothers about mothering, > etc. There is apparently no reason to listen to anyone besides Our > Royal Self, because We Know Better. The interest in claims of mathematicians depends on the quality of > mathematics done by those mathematicians. When, once upon a time, Mathematica was a Queen, with no ambiguities > and no arbitrariness in its foundations and its results, the opinions > of those who admired and mastered that queen were considered valuable > and were highly appreciated. Since Mathematica has become a whore, receptive to every arbitrary > axiom and every form of nonsense like undefinable definitions, > finished infinities and so on, Mathematica's results and the opinions > of its adherents are of the same value, namely equal to that of the > youngest natural number. > All quite rhetorical. > Why > does A ==> B not imply -A ==> -B? Why? The simplest answer is: because it /isn't/ one of the rules of > formal logic that A ==> B implies ~A ==> ~B. What more need be said? First of all, *why* this is a rule of formal logic, of course. But it is not a rule of formal logic that A->B implies ~A->~B is not > a rule of formal logic: that is instead an observation one makes / > about/ the rules of logic. So I will take your question to be why > isn't A->B implies ~A->~B a rule of formal logic? Well, for one thing, given the /other/ rules of formal logic, if we > also include (A->B) implies (~A->~B), then A->B implies A <-> B; and > so the symbol -> would be redundant in our calculations. Also, it > would then follow from (A and B)->A that (A and ~A)<->A for all > statements A; i.e. all statements and their negations would be > provably false if we included that rule, independent of whatever > axioms one might choose. So that is one reason why A->B implies ~A->~B > is not a rule of formal logic. That explanation is reasonable, but an independent explanation would > have been to say that > A ==> B implies ~A ==> ~B > is wrong because it does not fit reality. There are counter examples > in reality. > But A->B is not a statement about reality. It is a series of symbols, just as ~A -> ~B is a series of symbols. Formal logic is about the results of valid manipulations of those symbols; that is the only reality it claims to refer to. a man, then it is simply a calculation which we perform to conclude > that we then agree: Socrates is mortal. And that corresponds to the reasoning process of most people. But you > generally do /not/ apply this sort of reasoning in your arguments; I apply just this kind of reasoning. For instance: If we agree that > the number of distinguishable paths in the binary tree increases from > level to level by just the number of nodes which are present in this > level, then the number of paths at any height of the tree cannot be > larger than the number of nodes above that position plus the number of > initial paths (which is 1). > And as has been repeated to you endlessly, the infinite tree is not associated with the sort of finite level you refer to with level by level in your argument; so your premise is not satisfied by the infinite tree, and therefore your conclusion does not follow for the infinite tree. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) All quite rhetorical. > But A->B is not a statement about reality. It is a series of > symbols, just as ~A -> ~B is a series of symbols. Formal logic is > about the results of valid manipulations of those symbols; that is the > only reality it claims to refer to. Like a nouveau riche who disowns his roots. Formal logic has been created by real men of reality and should stick to its roots. We see the nonsense emerging from forgetting them. I apply just this kind of reasoning. For instance: If we agree that > the number of distinguishable paths in the binary tree increases from > level to level by just the number of nodes which are present in this > level, then the number of paths at any height of the tree cannot be > larger than the number of nodes above that position plus the number of > initial paths (which is 1). And as has been repeated to you endlessly, quite rhetorical. Repetition is not sufficient, even endless repetition, if there is no valid argument. > the infinite tree is not > associated with the sort of finite level you refer to with level by > level in your argument; so your premise is not satisfied by the > infinite tree, and therefore your conclusion does not follow for the > infinite tree. Infinity is the name for this level by level argument. There is no other infinity. Dedekind gives an intuitive argument for the existence of an infinite set. Starting with an object t of one's thoughts, a thought-thing, one successively gets new thought-things by the thought of t, the thought of the thought of t, ...; hence the set of all thought-things of a human being is infinite. But of course the set can never have a cardinal number larger than every natural number. Same is true for the numbers which, according to Dedekind, are creations of the humans mind. It is impossible to create an actually infinite set in this way. You see, your image of infinity is nothing but a misunderstanding of ideas about infinity that you and your buddies cannot comprehend. My tree argument just covers the infinite tree. Nothing else would remain. Spill of your formalistic nonsense. Try to think. Contrary to the earlier efforts by Frege and Dedekind we arrive at the conviction that as a precondition for the possibility of scientific cognition certain illustrative imaginations and insights are required and logic alone is not sufficient. Operating with the infinite can only be secured by the finite. (D. Hilbert) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Formal logic has been > created by real men of reality and should stick to its roots. We see > the nonsense emerging from forgetting them. And most of that nonsense, at lest in this thread, has been produced by WM himself. His ignorance of and violation of the roots of logic destroy any credit he might otherwise deserve. I apply just this kind of reasoning. For instance: If we agree that > the number of distinguishable paths in the binary tree increases from > level to level by just the number of nodes which are present in this > level, then the number of paths at any height of the tree cannot be > larger than the number of nodes above that position plus the number of > initial paths (which is 1). And as has been repeated to you endlessly, quite rhetorical. Repetition is not sufficient, even endless > repetition, if there is no valid argument. WM rekes not his own rede. WM repeats, ad nauseam, his old hat false arguments despite being presented repeatedly with the evidences of their logical flaws. the infinite tree is not > associated with the sort of finite level you refer to with level by > level in your argument; so your premise is not satisfied by the > infinite tree, and therefore your conclusion does not follow for the > infinite tree. Infinity is the name for this level by level argument. There is no > other infinity. There is in mathematics and in mathematical set theory. That there is not in WM's anti-mathematics maunderings is irrelevant to mathematics, including its various set theories. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > And that corresponds to the reasoning process of most people. But you > generally do /not/ apply this sort of reasoning in your arguments; I apply just this kind of reasoning. For instance: If we agree that > the number of distinguishable paths in the binary tree increases That presumes that WM's definition of distinguishable paths must correspond to everyone else's definition of existing paths, but since WM's definition of distinguishable paths makes assumptions that are incompatible with existing paths, no one but WM is willing to make that agreement. And without that non-existent agreement, WM has no case. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) > ... > You are confused? Why should the sequence 10, 11, 12, ... not have a > first element? Do you think an apple, a prune and a lobster do not > have a first element (in this order) but an apple-th, a prune-th, a > lobster-th element? Apparently you now disagree with WM's statement that if the set of ordinals > had in order {0, 1, 2, ...} that 0 should be the zero-th element? If you say zero is an ordinal, than you has to start with a zero-th > element, since ordinals are numbers which points to elements in > orders. Again you are confusing a mathematical ordinal and a common language > ordinal. 1 is a mathematical ordinal but not a common language ordinal. > 1st is not a mathematical ordinal but it is a common language ordinal. This difference, you claim, exists only in your dreams. There is only a difference between set theoretic ordinals and usual ordinals. Your common language ordinals are the ordinals. The only senseful ordinals. Ordinals starts always with the first ordinal in numbering. But not > the first element of a sequence is its first ordinal. The first > ordinal is the first element of the sequence of _ordinals_ not the > first element of arbitrary sequences. > You get more and more confused, I think. No it is you, switching from mathematical ordinals to common language > ordinals and back again. What is second + third? You are really funny. What a question. So you now agree that if the list of ordinals is {0, 1, 2, ...} that 0 is _If_ the list of ordinals would {0, 1, 2, ...} than 0 would be the > first ordinal. But it isn't. But it is. Your 0-th ordinal is in fact the 1-st ordinal? Very interesting. 1st, 2nd, 3rd and so on are surely not mathematical terms, although they > are used on occasion, but not well-defined. In the same way 0th is used > as in the 0th power of this expression. But you are confusing common > language ordinal numbers with mathematical ordinal numbers. The > mathematical ordinal number does *not* point to something in a sequence, > but it is an order type. And so perhaps not a number in the way you > would expect a number. But that sort of thing also occurs in common > language, a red herring is not a herring and also not red. You are only speaking of ordinals in the set theoretic manner. Yes, I know no other definition in mathematics. That's the problem. Maybe the books you are using are uncomplete? I'm > speaking of the ordinal concept of numbers, from which the set > theoretic concept may be derived. You are using the common language terminology for ordinal. That is > *not* the mathematical terminology. 1 is a mathematical ordinal > number but not a common language ordinal number. In common language you have to distinguish between ordianl and cardinal, that's no problem. If there is one apple, there is only one apple. If there is the first apple, maybe there are more apples. But one and first bases both on 1. Easyly to see here: X one, consists of only a first element XX two, consists of a first and a second element XXX three, consists of a first, a second and a third element ... But numbers are before sets. Mathematics is at first the teachings of > the numbers, not the teachings of the sets. Perhaps, and what is the problem here? No problem of mine. But a problem of set theory, maybe? So, if a set theoretic concept conflicts with the properties of the > naturals, the set theoretic concept fails in being universal. *What* property of naturals? And can you *prove* that property? > -- > 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: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > If you say zero is an ordinal, than you has to start with a zero-th > element, since ordinals are numbers which points to elements in > orders. > > Again you are confusing a mathematical ordinal and a common language > ordinal. 1 is a mathematical ordinal but not a common language ordinal. > 1st is not a mathematical ordinal but it is a common language ordinal. > > This difference, you claim, exists only in your dreams. No. > There is only a difference between set theoretic ordinals and usual > ordinals. The set theoretic ordinals are the only ordinals used in mathematics... > The only > senseful ordinals. Just opinion. > Ordinals starts always with the first ordinal in numbering. But not > the first element of a sequence is its first ordinal. The first > ordinal is the first element of the sequence of _ordinals_ not the > first element of arbitrary sequences. > You get more and more confused, I think. > > No it is you, switching from mathematical ordinals to common language > ordinals and back again. What is second + third? > > You are really funny. What a question. Well, for mathematical ordinals when you add the ordinal 2 to the ordinal 3 you get 5. > _If_ the list of ordinals would {0, 1, 2, ...} than 0 would be the > first ordinal. But it isn't. > > But it is. > > Your 0-th ordinal is in fact the 1-st ordinal? Very interesting. 0-th and 1-st are not mathematical terms. You are using common language there and 0 is the 1-st mathematical ordinal. > You are only speaking of ordinals in the set theoretic manner. > > Yes, I know no other definition in mathematics. > > That's the problem. Maybe the books you are using are uncomplete? Care to point me to a book on mathematics that contains another definition of ordinal number in mathematics? > I'm > speaking of the ordinal concept of numbers, from which the set > theoretic concept may be derived. > > You are using the common language terminology for ordinal. That is > *not* the mathematical terminology. 1 is a mathematical ordinal > number but not a common language ordinal number. > > In common language you have to distinguish between ordianl and > cardinal, that's no problem. If there is one apple, there is only one > apple. If there is the first apple, maybe there are more apples. > But one and first bases both on 1. Perhaps, but that is irrelevant. 1 is a cardinal number and 1-st an ordinal number in common language. > But numbers are before sets. Mathematics is at first the teachings of > the numbers, not the teachings of the sets. > > Perhaps, and what is the problem here? > > No problem of mine. But a problem of set theory, maybe? What problem? -- 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: A consideration concerning the diagonal argument of G. Cantor posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q rv:1.7.13) Gecko/20060414,gzip(gfe),gzip(gfe) And so, if we talk about the quantity of collections, the difference > between a collection of 5 elements and a collection of 6 elements is > the quantity - and not anything other. But you *weren't* talking about quantity. You said: Natural numbers can not be invented. If there is a collection of five > apples and a collection of six apples, what is the difference between > this collections? How am I supposed that this last statement, thrown at me out of the > blue, should have anything to do with the preceding sentence? This is > especially true, given that there is a *mathematical* definition for > the difference of two sets A and B, frequently written as AB. --- Albrecht S. Storz === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > And æby whose authority is it the complete list? That's always my question too. Since the lists in Cantor's proof are all provably incomplete, the alleged proof is never complete. Good observation! === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > And æby whose authority is it the complete list? That's always my question too. Since the lists in Cantor's proof are all provably incomplete, the alleged proof is never complete. On the contrary, the incompleteness of those lists is the whole point. Once that is established, the proof is complete, BECAUSE none of the lists are complete. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) And æby whose authority is it the complete list? That's always my question too. Since the lists in Cantor's proof are all provably incomplete, the alleged proof is never complete. On the contrary, the incompleteness of those lists is the whole point. > Once that is established, the proof is complete, BECAUSE none of the > lists are complete. One incompleteness implies the other incompletenes. For numbers that exist in the platonic sense it is uninteresting whether they are aligned in a single sequence before or after the diagonalization process. Only if during the diagonalization process a *new* number is created, then we have a new situation. It is appropriately called potential infinity. But that would spoil Cantor's diagonal argument, because his list is then also potentially infinite only and cannot not completely exist. Therefore, the diagonal argument is wrong under any circumstances. Too hard to understand? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) And æby whose authority is it the complete list? That's always my question too. Since the lists in Cantor's proof are all provably incomplete, the alleged proof is never complete. On the contrary, the incompleteness of those lists is the whole point. > Once that is established, the proof is complete, BECAUSE none of the > lists are complete. One incompleteness implies the other incompletenes. When establishing incompleteness of a list is the goal, as it is in the Cantor diagonal theorem, then establishing that incompleteness completes the proof. That those, like WM, who wish to deny the result, are therefore unwilling to understand the proof, does not invalidate that proof. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Pigeon hole principle. Which fails for infinite sets, as Hilbert's Hotel demonstrates Wrong observation. There are as many pigeons as holes (guests as rooms). If the principle would fail for infinite sets, then bijections would be useless for infinite sets. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor Pigeon hole principle. Which fails for infinite sets, as Hilbert's Hotel demonstrates Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. One definition (Dedekind's) of the infiniteness of a set is that it has an injection to a proper subset of itself, which directly contravenes any pigeonhole principle for such sets. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Pigeon hole principle. Which fails for infinite sets, as Hilbert's Hotel demonstrates Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. One definition (Dedekind's) of the infiniteness of a set is that it has > an injection to a proper subset of itself, which directly contravenes > any pigeonhole principle for such sets. There are more than one contradictions. For instance, according to Dedekind, infinity is only potential: Dedekind gives an intuitive argument for the existence of an infinite set. Starting with an object t of one's thoughts, a thought-thing, one successively gets new thought-things by the thought of t, the thought of the thought of t, ...; hence the set of all thought-things of a human being is infinite. But of course the set can never have a cardinal number larger than every natural number. Same is true for the numbers which, according to Dedekind, are creations of the humans mind. It is impossible to create an actually infinite set in this way. You see, your image of infinity is nothing but a misunderstanding of ideas about infinity that you and your buddies cannot comprehend. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Pigeon hole principle. Which fails for infinite sets, as Hilbert's Hotel demonstrates Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. One definition (Dedekind's) of the infiniteness of a set is that it has > an injection to a proper subset of itself, which directly contravenes > any pigeonhole principle for such sets. There are more than one contradictions. Apparently WM does not even know what the pigeonhole principle says: http://en.wikipedia.org/wiki/Pigeonhole_principal Pigeonhole principle ææ(Redirected from Pigeonhole principal) The inspiration for the name of the principle: pigeons in holes. Here næ=æ7 and mæ=æ9 so we can conclude that there are at least two empty pigeonholes. (There would be three if exactly two birds shared one hole.) The pigeonhole principle, also known as Dirichlet's box (or drawer) principle, states that, given two natural numbers n and m with n > m, if n items are put into m pigeonholes, then at least one pigeonhole must contain more than one item. Another way of stating this would be that m holes can hold at most m objects with one object to a hole; adding another object will force one to reuse one of the holes, provided that m is finite. More formally, the theorem states that there does not exist an injective function on finite sets whose codomain is smaller than its domain. Which clearly limits the PHP to finite sets only. So that WM's attempts to claim that the PHP applies to infinite sets is just another evidence of his mathematical and logical incompetence. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Pigeon hole principle. Which fails for infinite sets, as Hilbert's Hotel demonstrates Wrong observation. There are as many pigeons as holes (guests as > rooms). If the principle would fail for infinite sets, then bijections > would be useless for infinite sets. One definition (Dedekind's) of the infiniteness of a set is that it has > an injection to a proper subset of itself, which directly contravenes > any pigeonhole principle for such sets. There are more than one contradictions. Apparently WM does not even know what the pigeonhole principle says: http://en.wikipedia.org/wiki/Pigeonhole principal ææ(Redirected from Pigeonhole principal) The inspiration for the name of the principle: pigeons in holes. Here > næ=æ7 and mæ=æ9 so we can conclude that there are at least two empty > pigeonholes. (There would be three if exactly two birds shared one hole.) > The pigeonhole principle, also known as Dirichlet's box (or drawer) > principle, states that, given two natural numbers n and m with n > m, if > n items are put into m pigeonholes, then at least one pigeonhole must > contain more than one item. Another way of stating this would be that m > holes can hold at most m objects with one object to a hole; adding > another object will force one to reuse one of the holes, provided that m > is finite. More formally, the theorem states that there does not exist > an injective function on finite sets whose codomain is smaller than its > domain. > Which clearly limits the PHP to finite sets only. Of course. And all these finite sets are exhausted by case (1) below: For the minimal number A of symbols in at least one heap that is required to distinguish a maximal number B of heaps in a set of heaps we obtain following possibilities: 1) A = n in N <==> B = n in N. 2) A = n in N <==> B > n in N. 3) A > n in N <==> B = n in N. 4) A > n in N <==> B > n in N. Case (1) is obviously correct for finite heaps and sets. Cases (2) and (3) are impossible, because all natural numbers n are already absorbed by case (1). There is nothing left but case (4). === Subject: Re: WM begs another question > For the minimal number A of symbols in at least one heap that is > required to distinguish a maximal number B of heaps in a set of > heaps > we obtain following possibilities: 1) A = n in N <==> B = n in N. > 2) A = n in N <==> B > n in N. > 3) A > n in N <==> B = n in N. > 4) A > n in N <==> B > n in N. Since WM has not defined what he means by A > n in N or B > n in N, he cannot exclude any of 2, 3 or 4. Furthermore , WM's assumption that maximal numbers of symbols or maximal numbers of heaps need exist is begging the question, by assuming the very finiteness he is trying to prove. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > [...] So your way won't work in places my way will, >> e.g., where cardinalities have not been defined at all. >> It's an ODD approach, to say the least. [...] When Cantor started, there were no cardinalities defined, so that > while he could define 'Card(A)_<=_Card(B)' he did not yet have any > 'Card(A)' or 'Card(B)'. His definition of cardinalities came later. That is not correct. Cantor spoke of the power (M.8achtigkeit) -- his term for cardinality -- of a set from his earliest work on set theory and transfinite arithmetic. Granted, in his 1878 paper Ein Beitrag zur Mannigfaltigkeitslehre (his earliest on the topic, IIRC) he defined two collections to have the same power if they are equinumerous, which you might argue is just a definition of the binary predicate 'Card(A)_=_Card(B)'. I don't have access to that paper at the moment to see whether he refers independently to the power of this set or that, and he was of course still getting clear about all of this at the time (his early focus was on ordinals rather than cardinals). But by 1883 he was unquestionably referring to and quantifying over cardinals explicitly. Notably, refers to the powers of the number classes (classes of equinumerous ordinals) and proves in particular that the power of the class of countably infinite ordinals is uncountable, and he names these powers explicitly. He also claims there (without, as it happens, adequate justification) that every power is the power of some number class (this is of course equivalent to the axiom of choice); the quantification there is explicit, and is not elliptical for some statement about equinumerous classes. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > [...] So your way won't work in places my way will, >> e.g., where cardinalities have not been defined at all. > It's an ODD approach, to say the least. [...] When Cantor started, there were no cardinalities defined, so that > while he could define 'Card(A) <= Card(B)' he did not yet have any > 'Card(A)' or 'Card(B)'. His definition of cardinalities came later. That is not correct. Cantor spoke of the power (M.8achtigkeit) -- his > term for cardinality -- of a set from his earliest work on set theory > and transfinite arithmetic. In a letter to Dedekind of 20. June 1877 he announces his finding that lines, areas, and other continua of higher dimensionality have the same Maechtigkeit. > Granted, in his 1878 paper Ein Beitrag zur > Mannigfaltigkeitslehre (his earliest on the topic, IIRC) His earliest paper on that topic is in Crelles Journal f. Mathematik Bd. 77, S. 258 - 262 (1874), but there he does not yet use Maechtigkeit. es ist immer nur von einer 'eindeutigen Zuordnung' (one-to-one correspondence) der Elemente einer Gesamtheit zu denen einer anderen die Rede Zermelo notes. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) said: > [...] So your way won't work in places my way will, e.g., where > cardinalities have not been defined at all. > It's an ODD approach, to say the least. [...] >> When Cantor started, there were no cardinalities defined, so that >> while he could define 'Card(A)_<=_Card(B)' he did not yet have any >> 'Card(A)' or 'Card(B)'. His definition of cardinalities came >> later. >> That is not correct. Cantor spoke of the power (M.8achtigkeit) -- his >> term for cardinality -- of a set from his earliest work on set theory >> and transfinite arithmetic. In a letter to Dedekind of 20. June 1877 he announces his finding that > lines, areas, and other continua of higher dimensionality have the > same Maechtigkeit. > Granted, in his 1878 paper Ein Beitrag zur Mannigfaltigkeitslehre >> (his earliest on the topic, IIRC) His earliest paper on that topic is in Crelles Journal f. Mathematik > Bd. 77, S. 258 - 262 (1874), Right. I was talking about his earliest papers on pure set theory, of which I think the 1878 paper is reasonably considered the first. If I recall correctly, the 1874 paper is where he first proved that the reals are uncountable, but that the focus of the paper was still somewhat applied. I haven't actually looked at these papers for a number of years so could be misremembering. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > said: > [...] So your way won't work in places my way will, e.g., where > cardinalities have not been defined at all. >> It's an ODD approach, to say the least. [...] > When Cantor started, there were no cardinalities defined, so that >> while he could define 'Card(A) <= Card(B)' he did not yet have any >> 'Card(A)' or 'Card(B)'. æHis definition of cardinalities came >> later. > That is not correct. æCantor spoke of the power (M.8achtigkeit) -- his >> term for cardinality -- of a set from his earliest work on set theory >> and transfinite arithmetic. In a letter to Dedekind of 20. June 1877 he announces his finding that > lines, areas, and other continua of higher dimensionality have the > same Maechtigkeit. > æGranted, in his 1878 paper Ein Beitrag zur Mannigfaltigkeitslehre >> æ(his earliest on the topic, IIRC) His earliest paper on that topic is in Crelles Journal f. Mathematik > Bd. 77, S. 258 - 262 (1874), Right. æI was talking about his earliest papers on pure set theory, of > which I think the 1878 paper is reasonably considered the first. æIf I > recall correctly, the 1874 paper is where he first proved that the reals > are uncountable, but that the focus of the paper was still somewhat > applied. æ All of Cantor's work was applied. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > All of Cantor's work was applied. To what did Cantor apply his work showing that the cardinality of the reals exceeded the cardinality of the naturals? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > In a letter to Dedekind of 20. June 1877 he announces his finding that > lines, areas, and other continua of higher dimensionality have the > same Maechtigkeit. But not the same Maechtigkeit as the set of naturals. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world >>You see that this is nonsense (i.e. not a proper definition) from a >>logical point of view, right? >> I don't see any problem with it as a definition. Could you please >> explain why it's problematic to define a term to mean that there >> is an injection from one set to another? There's nothing wrong with defining a term such as is >dominated by to mean that there is an injection from >one set to another. The problem that both Balthasar -- >and especially MoeBlee -- have is when one tries to >define _cardinality_ this way. Okay, this has been quite an informative exchange that was sparked by my -- Michael F. Stemper #include Visualize whirled peas! === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 22, 10:06æam, mstem...@walkabout.empros.com (Michael Stemper) >You see that this is nonsense (i.e. not a proper definition) from a >>logical point of view, right? >> I don't see any problem with it as a definition. Could you please >> explain why it's problematic to define a term to mean that there >> is an injection from one set to another? There's nothing wrong with defining a term such as is >dominated by to mean that there is an injection from >one set to another. The problem that both Balthasar -- >and especially MoeBlee -- have is when one tries to >define cardinality this way. Okay, this has been quite an informative exchange that was sparked by my Except lwal's quote above, of what he claims I take to be the problem is not a correct paraphrase of anything I've posted. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > That is like saying that in order to define 'sin(x)',I have to be able > to eliminate æthe 'si' in 'sin'. That's a specious analogy. I'm surprised you resorted to it. At a formal level, 'sin' is understood to be an English phrase standing for a function symbol of the language. My objection is nothing related nor am I asking for demanding definition of anything finer than the actual constituent units of the language. And I'm not even arguing that in general a larger expression can't serve as a syntactical substitute. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But for my way there is no necessity for any set, A, æto have a > cardinality, Card(A), æat all, whereas your way requires the existence > of such cardinalities. So your way won't work in places my way will, > e.g., where cardinalities have not been defined at all. My point was not as to what is possible to work in certain places, but rather the mere question of elminability. But, I think I see how one might set your notation up so that it is eliminable. I'll post that later. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) But for my way there is no necessity for any set, A, æto have a > cardinality, Card(A), æat all, whereas your way requires the existence > of such cardinalities. So your way won't work in places my way will, > e.g., where cardinalities have not been defined at all. My point was not as to what is possible to work in certain places, but > rather the mere question of elminability. But, I think I see how one might set your notation up so that it is > eliminable. I'll post that later. To say that for sets A and B, 'Card(a)_<=_Card(B)' is defined to mean 'there exists an injection from A to B' certainly makes 'Card(a)_<=_Card(B)' eliminable, at least if there is no way regard any substring of it as being defined. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But for my way there is no necessity for any set, A, æto have a > cardinality, Card(A), æat all, whereas your way requires the existence > of such cardinalities. So your way won't work in places my way will, > e.g., where cardinalities have not been defined at all. My point was not as to what is possible to work in certain places, but > rather the mere question of elminability. But, I think I see how one might set your notation up so that it is > eliminable. I'll post that later. To say that for sets A and B, 'Card(a) <= Card(B)' is defined to mean > 'there exists an injection from A to B' certainly makes > 'Card(a) <= Card(B)' eliminable, at least if there is no way regard any > substring of it as being defined. Yes, I think I agree now. But there are a couple of considerations (which I don't claim contradict) eliminability. (1) Any expression that uses such a substring is one that STANDS for a well formed formula but is not itself exactly a well formed formula (since it has in it said substring that is not built from the recursive formula building rules), unless one were to revise the ordinary recursive rules defining 'well formed formula'. (2) I'm not sure that the explicit method for eliminability of such substrings is as straightforward as the very straightforward explicit method that comes right out of definitions in ordinary form. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> But, I think I see how one might set your notation up so that it is >> eliminable. I'll post that later. To say that for sets A and B, 'Card(a)_<=_Card(B)' is defined to mean > 'there exists an injection from A to B' certainly makes > 'Card(A)_<=_Card(B)' eliminable [...] > Sure. But WHY would anyone adopt such a strange predicate, instead of simply defining A << B :<-> there exists an injection from A into B ? Of course, later, after defining card(.), it turns out that Card(A) <= Card(B) <-> A << B. So what? B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > But, I think I see how one might set your notation up so that it is >> eliminable. I'll post that later. To say that for sets A and B, 'Card(a)_<=_Card(B)' is defined to mean > 'there exists an injection from A to B' certainly makes > 'Card(A)_<=_Card(B)' eliminable [...] Sure. But WHY would anyone adopt such a strange predicate, instead of > simply defining A << B :<-> there exists an injection from A into B Ask Cantor, as he was the one who did it. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > 1. What is a valid definition of cardinality that satisfies > the eliminability criterion, in ZF_C_? Any of the usual definitions will do, such as the one you described. > 2. Cab there even be a valid definition of cardinality that > satifies eliminability, that works in both ZFC and ZF+~AC? Using Scott's trick we can define card(A) to be {x | x is equipollent to A and for all y of rank < rank(x), y is not equipollent to x}, where a set x is equipollent to a set y if there exists a bijection between x and y. > 4. One often criticizes WM, TO, Albrecht, etc., for not > knowing the standard definition of cardinality, but how can > we expect them to know it if not even the standard > mathematicians can agree on a proper definition? The disagreement here has concerned utterly irrelevant details of the technical formal machinery. Not much need be made of it. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > Using Scott's trick we can define card(A) to be {x | x is equipollent > to A and for all y of rank < rank(x), y is not equipollent to x}, > where a set x is equipollent to a set y if there exists a bijection > between x and y. What is rank(x)?, And in ZF, does {x | x is equipollent to A and for all y of rank < rank(x), y is not equipollent to x} necessarily define a set? Doesn't one need to have, in ZF and ZFC, a set in which all the members satisfying a predicate must fall in order to have a predicate define a set. As in http://en.wikipedia.org/wiki/Axiom_schema_of_specification ? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > What is rank(x)?, The rank of the set x, defined by rank(x) = sup{ rank(y) | y in x } The rank of a set x is the least ordinal alpha such that x is in (alpha+1)-th level of the cumulative hierarchy. > And in ZF, does {x | x is equipollent to A and for all y of rank < > rank(x), y is not equipollent to x} necessarily define a set? Yes. Since A is equipollent to itself all members of {x | x is equipollent to A and for all y of rank < rank(x), y is not equipollent to x} are of rank <= rank(A), and the totality of such sets is provably a set -- in fact, it is just a level of the cumulative hierarchy. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) On 20 Jul 2008 18:49:31 +0300, Aatu Koskensilta said: > What is rank(x)?, The rank of the set x, defined by rank(x) = sup{ rank(y) | y in x } The rank of a set x is the least ordinal alpha such that x is in > (alpha+1)-th level of the cumulative hierarchy. > And in ZF, does {x | x is equipollent to A and for all y of rank < >> rank(x), y is not equipollent to x} necessarily define a set? Yes. Since A is equipollent to itself all members of {x | x is > equipollent to A and for all y of rank < rank(x), y is not equipollent > to x} are of rank <= rank(A), and the totality of such sets is > provably a set -- in fact, it is just a level of the cumulative > hierarchy. A subset of a level, no? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) What is rank(x)?, The rank of the set x, defined by rank(x) = sup{ rank(y) | y in x } That, as a definition of rank, looks remarkably circular. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) The rank of the set x, defined by rank(x) = sup{ rank(y) | y in x } > > That, as a definition of rank, looks remarkably circular. Consider the empty set 0. By the above we have rank(0) = sup { rank(y) | y in x} = sup {} = 0 grounding the definition. It is an example of definition by epsilon-recursion, the dual of epsilon-induction: If, whenever all members of a set X have the property X, X also has the property P, all sets have the property P. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > The rank of the set x, defined by rank(x) = sup{ rank(y) | y in x } > > That, as a definition of rank, looks remarkably circular. Consider the empty set 0. By the above we have rank(0) = sup { rank(y) | y in x} = sup {} = 0 I fail to see that something requiring an a priori meaning for rank(y) | y in x defines rank(0). What sort of objects are such rankings supposedly limited to being, or can any set be a rank of some suitable set? Furthermore, sup is to be taken with respect to what sort of ordering? There are at least two such orderings in ZF, membership and subset, which do not coincide. While the y in x would suggest membership ordering, if that is the case, it's being the case should be made more clearly. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > The rank of the set x, defined by >> rank(x) = sup{ rank(y) | y in x } >> >> That, as a definition of rank, looks remarkably circular. >> Consider the empty set 0. By the above we have >> rank(0) = sup { rank(y) | y in x} = sup {} = 0 I fail to see that something requiring an a priori meaning for > rank(y) | y in x defines rank(0). Typo; he meant: rank(0) = sup { rank(y) | y in 0} = sup {} = 0 > What sort of objects are such rankings supposedly limited to being, They are ordinals. > or can any set be a rank of some suitable set? Furthermore, sup is to be taken with respect to what sort of > ordering? less-than on the ordinals, i.e., membership. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > The rank of the set x, defined by >> rank(x) = sup{ rank(y) | y in x } >> >> That, as a definition of rank, looks remarkably circular. >> Consider the empty set 0. By the above we have >> rank(0) = sup { rank(y) | y in x} = sup {} = 0 I fail to see that something requiring an a priori meaning for > rank(y) | y in x defines rank(0). Typo; he meant: rank(0) = sup { rank(y) | y in 0} = sup {} = 0 What sort of objects are such rankings supposedly limited to being, They are ordinals. or can any set be a rank of some suitable set? Furthermore, sup is to be taken with respect to what sort of > ordering? less-than on the ordinals, i.e., membership. For the von Neumann ordinals, isn't being a member equivalent to being a proper subset? Is it different for other forms or ordinals? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > For the von Neumann ordinals, isn't being a member equivalent to being a > proper subset? Yes. However, the rank function is a function from arbitrary sets to ordianals, defined by recursion on the membership relation, not on the less-than -relation for ordinals. The principle of definition by epsilon-recursion says that given an operation G: V --> V there is a unique operation F: V --> V such that F(x) = G({ | y in x}) Say a (set-sized) function f is an approximation of F if the domain of f is transitive and f satisfies the above equation. By epsilon-induction we can show that for any set A there is an approximation of F with A in its domain, and that all such approximations agree. We thus obtain an explicit definition of F F(x) = y <==> for every approximations f of F with x in its domain, f(x) = y In case of the rank function, the operation G is simply G(x) = sup {z u {z} | in x for some y} corresponding to the equation rank(x) = sup { rank(y) + 1 | y in x} -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <87mykc7luv.fsf@alatheia.dsl.inet.fi> <87bq0s4ktw.fsf@alatheia.dsl.inet.fi> <87y73w33pp.fsf@alatheia.dsl.inet.fi> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > The rank of the set x, defined by > ærank(x) = sup{ rank(y) | y in x } > That, as a definition of rank, looks remarkably circular. > Consider the empty set 0. By the above we have > ærank(0) = sup { rank(y) | y in x} = sup {} = 0 > I fail to see that something requiring an a priori meaning for æ > rank(y) | y in x defines rank(0). Although Aatu and Virgil agree on the definition of cardinality, they disagree on the definition of rank. I think both of them make valid points. On one hand, I've seen many recursive definitions of the type that Aatu has given. And so I see nothing wrong with Aatu's definition of rank to find rank(0). But Virgil is right in that this definition doesn't work -- and to see why, let's find rank({0}): > rank(x) = sup{ rank(y) | y in x } So rank({0}) = sup{ rank(y) | y in {0}} = sup{ rank(0) } = sup{0} = 0 And indeed, we discover that according to Aatu's definition, the rank of every set is zero! What Aatu probably meant is something like: rank(x) = sup{ rank(y)+1 | y in x } Then rank(0) is still zero (since it's still sup{}), but now rank({0}) is one, and (if I did it correctly) the rank of every von Neumann ordinal should be itself. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > The rank of the set x, defined by > ærank(x) = sup{ rank(y) | y in x } > That, as a definition of rank, looks remarkably circular. > Consider the empty set 0. By the above we have > ærank(0) = sup { rank(y) | y in x} = sup {} = 0 > I fail to see that something requiring an a priori meaning for æ > rank(y) | y in x defines rank(0). Although Aatu and Virgil agree on the definition of > cardinality, they disagree on the definition of rank. I think both of them make valid points. On one hand, I've seen many recursive definitions of > the type that Aatu has given. And so I see nothing > wrong with Aatu's definition of rank to find rank(0). But Virgil is right in that this definition doesn't > work -- and to see why, let's find rank({0}): rank(x) = sup{ rank(y) | y in x } So rank({0}) = sup{ rank(y) | y in {0}} > = sup{ rank(0) } > = sup{0} > = 0 And indeed, we discover that according to Aatu's > definition, the rank of _every_ set is zero! No. sup{0} = sup{{}} = U{{}} = {x | x in {{}} } = {{}} = 1. -- Michael Press === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) The rank of the set x, defined by > ærank(x) = sup{ rank(y) | y in x } > That, as a definition of rank, looks remarkably circular. > Consider the empty set 0. By the above we have > ærank(0) = sup { rank(y) | y in x} = sup {} = 0 > I fail to see that something requiring an a priori meaning for æ > rank(y) | y in x defines rank(0). Although Aatu and Virgil agree on the definition of > cardinality, they disagree on the definition of rank. I think both of them make valid points. On one hand, I've seen many recursive definitions of > the type that Aatu has given. And so I see nothing > wrong with Aatu's definition of rank to find rank(0). But Virgil is right in that this definition doesn't > work -- and to see why, let's find rank({0}): rank(x) = sup{ rank(y) | y in x } So rank({0}) = sup{ rank(y) | y in {0}} > = sup{ rank(0) } > = sup{0} > = 0 And indeed, we discover that according to Aatu's > definition, the rank of _every_ set is zero! No. > sup{0} = sup{{}} = U{{}} = {x | x in {{}} } = {{}} = 1. If U(A) means the union of A, as in ZF's axiom of union, then U({{}}) = {} = 0. And if it does not mean union in that sense, what does it mean? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) said: >> ... >> But Virgil is right in that this definition doesn't >> work -- and to see why, let's find rank({0}): >> rank(x) = sup{ rank(y) | y in x } >> So rank({0}) = sup{ rank(y) | y in {0}} >> = sup{ rank(0) } >> = sup{0} >> = 0 >> And indeed, we discover that according to Aatu's >> definition, the rank of _every_ set is zero! No. > sup{0} = sup{{}} = U{{}} = {x | x in {{}} } = {{}} = 1. Uhuh. By your definition UA = A. UA = {x | for some y in A, x in y}. Thus: sup{0} = sup{{}} = U{{}} = {x | for some y in {{}}, x in y } = {x | x in {}} = {} = 0. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) said: > ... > Although Aatu and Virgil agree on the definition of cardinality, they > disagree on the definition of rank. I think both of them make valid points. On one hand, I've seen many recursive definitions of the type that > Aatu has given. And so I see nothing wrong with Aatu's definition of > rank to find rank(0). But Virgil is right in that this definition doesn't work -- and to see > why, let's find rank({0}): > rank(x) = sup{ rank(y) | y in x } So rank({0}) = sup{ rank(y) | y in {0}} > = sup{ rank(0) } > = sup{0} > = 0 And indeed, we discover that according to Aatu's definition, the rank > of _every_ set is zero! What Aatu probably meant is something like: rank(x) = sup{ rank(y)+1 | y in x } I suspect Aatu simply meant by sup(A) the least *strict* upper bound of A, for sets of ordinals A. (That's how I interpreted him, but you're right that sup(A) is usually just taken to be the union of A.) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) >> The rank of the set x, defined by >> rank(x) = sup{ rank(y) | y in x } > > That, as a definition of rank, looks remarkably circular. >> Consider the empty set 0. By the above we have >> rank(0) = sup { rank(y) | y in x} = sup {} = 0 I fail to see that something requiring an a priori meaning for > rank(y) | y in x defines rank(0). The justification of the recursive definition of rank (e.g. how you can eliminate it) requires transfinite recursion. Textbooks implicitly assume this and rarely if ever show the direct definition. (Or they define it in terms of the cumulative hierarchy, which in turn will have a recursive definition.) While I'm not sure if Metamath's approach will help, it uses a hierarchy of only direct definitions, so that you can see what the direct definition would be if expanded out (it is extremely long and tedious, which explains why textbooks avoid it). Specifically, we have http://us.metamath.org/mpegif/df-rdg.html (recursive def. generator) -> http://us.metamath.org/mpegif/df-r1.html (cum. hierarch. of sets) -> http://us.metamath.org/mpegif/df-rank.html (rank). The recursive definition is then proved as a theorem: http://us.metamath.org/mpegif/rankval4.html > What sort of objects are such rankings supposedly limited to being, or > can any set be a rank of some suitable set? A rank is an ordinal number, and the rank function's domain is all sets. > Furthermore, sup is to be taken with respect to what sort of ordering? > There are at least two such orderings in ZF, membership and subset, > which do not coincide. I think sup here is supposed to be simply set union. > While the y in x would suggest membership ordering, if that is the > case, it's being the case should be made more clearly. I think Aatu's definition should be rank(x) = sup{ rank(y)+1 | y in x } where +1 is ordinal successor, unless I'm misinterpreting sup. -- Norm http://us.metamath.org/email.html (Reply to author at this URL. The from address in this post is not valid.) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) It is only æthe complete statement which has > been defined, and there is no reason to suppose that any proper part of > it, other than A and B representing sets, is required to have any > meaning. No, that does not work in a fully rigorous manner. It does not satisfy > the conditions of eliminability and non-creativity. Sure it does. A definition of this sort provides an explicit method for transforming a statement containing the defined expression into one that does not. Such definitions are of course not unproblematic, from a purely pragmatic point of view: it is not apparent from the syntactic structure of a statement involving contextually defined expressions what inferences are licensed. Indeed, in the system of _Principia Mathematica_ we find many a contextual definition that easily leads to confusion. As an example, one can produce a seemingly correct proof of the axiom of reducibility which, when the defined expressions are expanded, turns out to be invalid. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <87r69o7m7x.fsf@alatheia.dsl.inet.fi> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 20, 5:51æam, Aatu Koskensilta pragmatic point of view: I had agreed to that from the outset. > it is not apparent from the syntactic > structure of a statement involving contextually defined expressions > what inferences are licensed. Indeed, in the system of Principia > Mathematica we find many a contextual definition that easily leads to > confusion. As an example, one can produce a seemingly correct proof of > the axiom of reducibility which, when the defined expressions are > expanded, turns out to be invalid. That is interesting. Does that problem concern contextual defintion of the iota operator? Anyway, it has long been a question of mine how one would give a rigorously RECURSIVE method of converting a PM formula (of arbitrary complexity of nested formulas and nested definition descriptions, etc.) into a primitive formula without the iota operator. If one could do that, then that would obviate having to use double induction and simultaneous recursion to fully formalize the iota operator as a primitve, as instead we could have it defined and elminable with full rigor that is not apparent just from Russell's remarks (including his order of occurence plan, which does not account for arbitrary complexity of nesting). But I backed off using the word 'contextual definition' in this discussion, since I'm not sure that Virgil's example should be described that way. If one considers such questions or problems to be pointlessly pedantic, then one can ignore them, or perhaps consider them as if asked in a context where (for whatever reason) we were trying to get all of this defined notation to be processed by a computer program for checking syntax and things like that. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <87r69o7m7x.fsf@alatheia.dsl.inet.fi> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 20, 5:51 am, Aatu Koskensilta It is only the complete statement which has > been defined, and there is no reason to suppose that any proper part of > it, other than A and B representing sets, is required to have any > meaning. No, that does not work in a fully rigorous manner. It does not satisfy > the conditions of eliminability and non-creativity. Sure it does. A definition of this sort provides an explicit method > for transforming a statement containing the defined expression into > one that does not. Ah, what dawned on me just after I finished posting yesterday is that I think I see that one may provide an explicit method after all. So I think my remark to the contrary, taken at face value, is incorrect. However, I'm still sorting through what differences there may be in an explicit method for definitions in ordinary form and for those in Virgils's form. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) On 20 Jul 2008 15:51:14 +0300, Aatu Koskensilta >> No, that does not work in a fully rigorous manner. It does not satisfy >> the conditions of eliminability and non-creativity. > Sure it does. A definition of this sort provides an explicit method > for transforming a statement containing the defined expression into > one that does not. > Right, but... if we later, after defining /card(.) <= card(.)/ would introduce /card/ and /<=/ in isolation (i.e. as defined terms in their own right) we clearly would have a problem from a strictly formal point of view. B. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > On 20 Jul 2008 15:51:14 +0300, Aatu Koskensilta > No, that does not work in a fully rigorous manner. It does not satisfy >> the conditions of eliminability and non-creativity. > Sure it does. A definition of this sort provides an explicit method > for transforming a statement containing the defined expression into > one that does not. Right, but... if we later, after defining /card(.) <= card(.)/ would > introduce /card/ and /<=/ in isolation (i.e. as defined terms in their > own right) we clearly would have a problem from a strictly formal point > of view. If these new definitions were, as they should be, entirely compatible with the old 'Card(.) <= Card(.)', one could simply retire the older one, and replace it everywhere with the newer ones, by showing that for all A, B, Old[Card(A) <= Card(B)] <==> New[Card(A) <= Card(B)] === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) The above definition is not intended as a definition of cardinality, but > only a definition of the statement 'Card(A) <= Card(B)'. And a fully rigorous first order system does not allow definitions of > statements in that manner. Quite. Nevertheless, we do, in mathematics, and set theory in particular, meet definitions of this sort. It is but pointless pedantry to dwell on the details of this or that treatment of definitions in first-order logic on this occasion. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Here, let me try your middle ground method: > Poster A says, 2+2=4, 2+1=3 and 0+1=1 are all true statements of the > arithmetic of the natural numbers. > Poster B says, 2+2=4, 2+1=0 (as a compromise, to have some > alternatives) æand 0+1=1 are all true statements of the arithmetic of > the natural numbers. > Poster C says, 2+2=1, 2+1=0 and 0+1=5 are all true statements of the > arithmetic of the natural numbers. > So Poster B has the best approach since his is a middle ground > between the standard arithmetic of Poster A and the unstandard radical > alternative of Poster C. > Yeah, the middle-ground philosophy in action. Can't beat it! This is an excellent point -- indeed, I've been waiting for someone to make this comparison. Suppose there really was a thread here at sci.math where there was someone like Poster C. Naturally, the other posters would refer to Poster C as a crank, etc. And indeed, to the standard mathematicians, there is little difference between a formula such as 2+1 = 0 and, say, card(N) = card(R) -- after all, the negation of 2+1 = 0 is a theorem of ZFC, just as the negation of card(N) = card(R) is a theorem of ZFC. Now the question is, would I try to defend Poster C? Or if someone made a middle ground like Poster B, would I defend Poster B? My response is that I wouldn't -- and here's why. For one thing, there aren't very many posters like B or C. Very few posters at sci.math challenge theorems of arithmetic or finite set theory. Only infinite set theory is being challenged in most threads. And if someone were to make a post like B or C, certainly it wouldn't result in a thread of over 8000 posts. Challenges to arithmetic or finite set theory aren't very interesting, whereas challenges to infinite set theory are. I find the debate on infinite set theory to be interesting -- and judging by the length of the thread, many others find the debate interesting as well. The closest that anyone has ever come to being a poster like B or C is Anthony Aiya-Oba, who once claimed back in late 2007 that 1^2 = 3, in a thread that lasted 65 posts. Did I defend Aiya-Oba, or a middle-ground poster like our hypothetical Poster B? I didn't defend him -- indeed, I didn't even post in the thread, because I don't find 1^2 = 3 or other challenges to arithmetic or finite set theory interesting. If someone can come up with a theory with a formula such as 1^2 = 3 or 2+1 = 0, and looks like an interesting (in my own personal opinion) theory to defend, then I may actually try to defend them. But until then, I'd probably just ignore posts similar to Poster B or C. BTW, I actually have defended theories with a nonstandard version of signed numbers -- most notably, Timothy Golden's polysigned numbers. But I have yet to defend a theory that differs from the standard theories of PA or ZFC when it comes to the natural numbers -- and I probably never will. The theories that I find are worth debating or taking a middle ground position are infinite set theories. In particular, I believe that there can be rigorous (and not trivially inconsistent) set theories in which some (not all, but some ) of the claims that the opponents of ZFC have are provable. And I hope that through these debates, I may discover them. And I find the debate between Virgil and the other standard mathematicians regarding the definition of cardinality to be interesting, so I want to participate. And in this debate, I wish to take the middle ground position between full mathematical rigor and an informal approach. For full mathematical rigor can be a bit pedantic, but the informal approach often abuses notation. So I fall somewhere between pedantry and abuse of notation, for each have their uses. I admit that I was wrong to consider Balthasar to represent the middle ground position. For Balthasar, as he points out in his posts, is actually an adherent of NF(U), not ZF(C) -- while both MoeBlee and Virgil are using ZF(C). Balthasar's definition of cardinality doesn't even make sense in ZF(C), because Balthasar's (based on Frege's) cardinals would end up being proper classes, not sets. In the end, I will continue to read the posts in the subthread about the definition of cardinality, but I won't post there. And maybe galathaea really isn't the middle ground between standard mathematicians and the cranks. But once again, I agree with both sides in part. The standard mathematicians are correct to say that WM, etc., don't (yet) have rigorous theories. But still, their intuitions about infinite set theory aren't without merit. Perhaps with a little imagination and a little math, one can come up with set theories in which some of their intuitions are provable. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Here, let me try your middle ground method: > Poster A says, 2+2=4, 2+1=3 and 0+1=1 are all true statements of the > arithmetic of the natural numbers. > Poster B says, 2+2=4, 2+1=0 (as a compromise, to have some > alternatives) æand 0+1=1 are all true statements of the arithmetic of > the natural numbers. > Poster C says, 2+2=1, 2+1=0 and 0+1=5 are all true statements of the > arithmetic of the natural numbers. > So Poster B has the best approach since his is a middle ground > between the standard arithmetic of Poster A and the unstandard radical > alternative of Poster C. > Yeah, the middle-ground philosophy in action. Can't beat it! This is an excellent point -- indeed, I've been waiting for someone > to make this comparison. Suppose there really was a thread here at sci.math where there > was someone like Poster C. Naturally, the other posters would > refer to Poster C as a crank, etc. And indeed, to the standard > mathematicians, there is little difference between a formula > such as 2+1 = 0 and, say, card(N) = card(R) -- after all, the > negation of 2+1 = 0 is a theorem of ZFC, just as the negation of > card(N) = card(R) is a theorem of ZFC. No, actually I chose the example of 2+1 = 0 (and in context of the standard model) so to involve only finitistic statements; and not infinitistic ones such as you just mentioned. > Now the question is, would I try to defend Poster C? Or if someone > made a middle ground like Poster B, would I defend Poster B? My response is that I wouldn't -- and here's why. For one thing, > there aren't very many posters like B or C. Very few posters at > sci.math challenge theorems of arithmetic or finite set theory. That is aside the point. Indeed, I chose a finitistic example just to make the example as stark as possible. > Only > infinite set theory is being challenged in most threads. And if > someone were to make a post like B or C, certainly it wouldn't > result in a thread of over 8000 posts. Challenges to arithmetic or > finite set theory aren't very interesting, whereas challenges to > infinite set theory are. I find the debate on infinite set theory to > be interesting -- and judging by the length of the thread, many > others find the debate interesting as well. That is all aside the point. You made your choice of option B regarding a question not about infinities but rather about the matter of eliminability in definitions. > If someone can come up with a theory with a formula such as > 1^2 = 3 or 2+1 = 0, and looks like an interesting (in my > own personal opinion) theory to defend, then I may actually try > to defend them. But until then, I'd probably just ignore posts > similar to Poster B or C. No, I didn't ask about THEORIES. I specifically qualified that the statements were to be understood as about the standard model. I.e., just plain old finitistic arithmetic, not tied to any particular theory. > And I find the debate between Virgil and the other standard > mathematicians regarding the definition of cardinality to be > interesting, I'm not debating about what should be the definition of cardinality itself. I'm just talking about whether a particular definition (of whatever it is) satisfies the criterion of eliminability. > so I want to participate. And in this debate, I > wish to take the middle ground position between full > mathematical rigor and an informal approach. Those are NOT the two opposite grounds, at least not as far as my involvement. I'm not saying that one shouldn't relax rigor to get on with communicating mathematics. The matter I'm discussing is simply the exact technical matter of whether a certain formulation is or is not eliminable. > For full > mathematical rigor can be a bit pedantic, but the informal > approach often abuses notation. Yes, so? No one here is disputing that. > So I fall somewhere between pedantry and abuse of notation, > for each have their uses. No, pedanticism and egregious mis-notation are not the two sides of any argument here. Rather, the argument, or at least my own part in it, is as to whether a given formulation is or is not eliminable. MoeBlee === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > But once again, I agree with both sides in part. The standard > mathematicians are correct to say that WM, etc., don't (yet) have > rigorous theories. But still, their intuitions about infinite set > theory aren't without merit. What merit do you see in these intuitions? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) <87vdz06507.fsf@alatheia.dsl.inet.fi> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On 20 Jul., 15:48, Aatu Koskensilta mathematicians are correct to say that WM, etc., don't (yet) have > rigorous theories. But still, their intuitions about infinite set > theory aren't without merit. What merit do you see in these intuitions? What merit do you see in the dream of infinity? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > On 20 Jul., 15:48, Aatu Koskensilta mathematicians are correct to say that WM, etc., don't (yet) have > rigorous theories. But still, their intuitions about infinite set > theory aren't without merit. What merit do you see in these intuitions? What merit do you see in the dream of infinity? Working out the consequences of axiom systems is good mental training in logic, as well as being occasionally useful to those who choose not to exercise their minds in this way. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > What merit do you see in the dream of infinity? It is an intellectually appealing dream of great mathematical utility. In contrast, your intuitions and finite dreams seem not to lead to any mathematical developments of interest. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) > What merit do you see in the dream of infinity? It is an intellectually appealing dream of great mathematical > utility. In contrast, your intuitions and finite dreams seem not to > lead to any mathematical developments of interest. > The combination of infinite and finite dreams would lead to even a much more appealing utility: mathematical relativity! === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) And maybe galathaea really isn't the middle ground > between standard mathematicians and the cranks. But > once again, I agree with both sides in part. The standard > mathematicians are correct to say that WM, etc., don't > (yet) have rigorous theories. But still, their intuitions about > infinite set theory aren't without merit. Perhaps with a little > imagination and a little math, one can come up with set > theories in which some of their intuitions are provable. I appreciate your balanced position, but I do not share it. Mathematics, as a means to understand the dissimilarities of reality (Dedekind), does not need axioms or theories. Even theoretical proofs are less important than is always assumed. Some natural numbers and some lines in sand will do it as a starting point. The decisive proofs are obtained by experiments, possibly done with powerful computers. If you have an abacus, then you can already prove that I + I = II with much more rigor than any theoretical proof can supply. Everything that does not fit into this scheme, in particular every infinity, does not belong to mathematics but is idle speculation. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world > And maybe galathaea really isn't the middle ground > between standard mathematicians and the cranks. But > once again, I agree with both sides in part. The standard > mathematicians are correct to say that WM, etc., don't > (yet) have rigorous theories. But still, their intuitions about > infinite set theory aren't without merit. Perhaps with a little > imagination and a little math, one can come up with set > theories in which some of their intuitions are provable. I appreciate your balanced position, but I do not share it. > Mathematics, as a means to understand the dissimilarities of reality > (Dedekind), does not need axioms or theories. Even theoretical proofs > are less important than is always assumed. Some natural numbers and > some lines in sand will do it as a starting point. The decisive proofs > are obtained by experiments, possibly done with powerful computers. If > you have an abacus, then you can already prove that I + I = II with > much more rigor than any theoretical proof can supply. Everything that > does not fit into this scheme, in particular every infinity, does not > belong to mathematics but is idle speculation. Fortunately for mathematics, it is beyond the power of such misguided anti-mathematicians as WM to declare what mathematics is, or even what it should be. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Sorry, actually forgot actual: Everything that does not fit into this scheme, in particular every actual infinity, does not belong to mathematics but is idle speculation. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor(was Q) Distribution: world Sorry, actually forgot actual: Everything that does not fit into this scheme, in particular every > actual infinity, does not > belong to mathematics but is idle speculation. Mathematics is the art of idle speculation. === Subject: Number with Gauss, prime counting. Hello teacher~ The prime counting function pi(n), which counts the number of primes less than some integer n. When only 15 years old, Gauss proposed that pi(n) ~ [n / ln(n)]. n = 10^3 ==> pi(n) = 168 , [n / ln(n)] = 144.8 n = 10^4 ==> pi(n) = 1229 , [n / ln(n)] = 1085.7 n = 10^5 ==> pi(n) = 9592 , [n / ln(n)] = 8685.9 n = 10^6 ==> pi(n) = 78498 , [n / ln(n)] = 72382.4 ... --------------------------------------------------------- How did Gauss conjecture it ? Gauss was a genius ? === Subject: Re: Number with Gauss, prime counting. reply-type=response > Hello teacher~ The prime counting function pi(n), which counts the number of primes > less than some integer n. When only 15 years old, > Gauss proposed that pi(n) ~ [n / ln(n)]. n = 10^3 ==> pi(n) = 168 , [n / ln(n)] = 144.8 > n = 10^4 ==> pi(n) = 1229 , [n / ln(n)] = 1085.7 > n = 10^5 ==> pi(n) = 9592 , [n / ln(n)] = 8685.9 > n = 10^6 ==> pi(n) = 78498 , [n / ln(n)] = 72382.4 > ... --------------------------------------------------------- > How did Gauss conjecture it ? Probably heuristically. > Gauss was a genius ? Of course. === Subject: Half of SR and GR correct posting-account=NzomCgoAAADSxo8dItZFimQE_f4Fbqcn .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) Relative motion and relative acceleration are wrong. But the other half of the content of these theories is correct. It all works with absolute motion. Acceleration through space creates new motion. Every object can change motion in 3 ways. They can accelerate through space decelerate or change direction. === Subject: Minimum sub-sequence sum posting-account=zK-EhAoAAADx9fogrgJdoCe-9maW3qPu Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) I have a sequence of integers (positive/negative values) and i would like to find the sub sequence of consecutive values that yields the smallest sum, I can think of a very simple O(n^3) complex solution, however I was wondering are there any better solutions? and what is the name of this kind of problem? -Mich === Subject: Re: Minimum sub-sequence sum > I have a sequence of integers (positive/negative values) and i would > like to find the sub sequence of consecutive values that yields the > smallest sum, I can think of a very simple O(n^3) complex solution, > however I was wondering are there any better solutions? and what is > the name of this kind of problem? Call your integers a_1, a_2, ..., a_n. Compute the numbers s_1 = a_1, s_2 = a_1 + a_2, ... s_n = a_1 + a_2 + ... + a_n. That much should only take O(n) operations. Now calculate all the numbers s_i - s_j with i > j, and keep track of which of them is the smallest. That's O(n^2), and you're done. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Hyperbolic Geometry book recommendation please posting-account=AFsgCgkAAAA3VOfxqn2cTB2LbLN3nbER Gecko/20070319,gzip(gfe),gzip(gfe) Can anyone suggest a good but very basic book to act as a text/reference book for a short basic course on Hyperbolic Geometry? The students concerned would not necessarily know a huge amount of math already, but they would be expected to be familiar with sinh cosh and tanh etc. Bill === Subject: Re: Hyperbolic Geometry book recommendation please > Can anyone suggest a good but very basic book to act as > a text/reference book for a short basic course on > Hyperbolic Geometry? The students concerned would not necessarily know > a huge amount of math already, but they would be expected > to be familiar with sinh cosh and tanh etc. > Bill Non-Euclidean Geometry, Wolfe === Subject: Re: statistics and set theory How unusual! A moron posting from mathforum.org! > === Subject: Re: statistics and set theory posting-account=Jz4DtgkAAAAZkdWvJAd__jMF7l1N5_1V Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > once again i have a good idea :) we can merge cardinality and statistics. using limits and infinitesimals can be inappropriate to do statistics. e.g. > what is the probability that two randomly chosen reals are equal ? You need to define how to choose a random real, through a suitable probability measure. The reals cannot all have the same probability, or even the same probability density. But assuming you have a continuous probability distribution, and the two are chosen independently, then the probability the two are the same is zero. and what is the probability that two randomly chosen integers are equal ? Again you need to define how to choose a random integer, through a suitable probability measure. The integers cannot all have the same probability. But assuming you have a discrete probability distribution, and the two are chosen independently, then the probability the two are the same is positive. the above probabilities are not equal since countable is different from uncountable. Indeed; one is zero and the other is positive. and they are not zero either since it is possible. Possible (but very unlikely) events can have a probability of zero. The rest is meaningless. we cannot say 1/h and 1/h^2 either. thus the solutions are : 1/ aleph_0 and 1/ aleph_1. devision is not allowed or defined in set theory , thus i will call them aleph_-1 and aleph_-2. 2^aleph_-1 = aleph_-2. i will call this cardinal statistics > tommy1729 === Subject: computational complexity of Euclid's algorithm Wikipedia states that the computational complexity of Euclid's algorithm (using subtraction rather than division) is 2^n where n is the number of digits. Is this correct? My experimentation* suggests that the number of subtractions required is asymptotic to n^2. Mark * I'll post my code and its output if anyone wants but I'd prefer not to clutter up Usenet with my abysmal C unnecessarily :-) === Subject: Re: computational complexity of Euclid's algorithm > Wikipedia subtraction rather than division) is 2^n where n is the number of digits. Is this correct? My experimentation* suggests that the number of subtractions required is > asymptotic to n^2. If you try to find the gcd of 1 and N using repeated subtraction you'll do N subtractions, which is roughly 10^n, no? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: computational complexity of Euclid's algorithm > Wikipedia states that the computational complexity of Euclid's algorithm (using >> subtraction rather than division) is 2^n where n is the number of digits. >> Is this correct? >> My experimentation* suggests that the number of subtractions required is >> asymptotic to n^2. If you try to find the gcd of 1 and N using repeated subtraction > you'll do N subtractions, which is roughly 10^n, no? Yes. But that will only occur on average one in 10^n times. I have assumed that N is fixed and with n digits and that the smaller number is uniformly distributed between 0 and N-1. It may well be that having run the experiment for 2^30 that the numbers are still too small to observe the asymptotic behaviour. === Subject: Re: computational complexity of Euclid's algorithm > Wikipedia >> states that the computational complexity of Euclid's algorithm (using >> subtraction rather than division) is 2^n where n is the number of digits. >> Is this correct? >> My experimentation* suggests that the number of subtractions required is >> asymptotic to n^2. If you try to find the gcd of 1 and N using repeated subtraction > you'll do N subtractions, which is roughly 10^n, no? Yes. But that will only occur on average one in 10^n times. If we're talking about worst-case computational complexity, one time in 10^n is more than enough. Isn't that what your source means? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: computational complexity of Euclid's algorithm >> Yes. But that will only occur on average one in 10^n times. If we're talking about worst-case computational complexity, > one time in 10^n is more than enough. Isn't that what your > source means? I think you are right. It says worst case in the previous paragraph when considering Euclid's algorithm with divisions rather than subtractions. Mark === Subject: Re: computational complexity of Euclid's algorithm posting-account=VR0DOgoAAADggPTteFeA2AkmHNhjcrDV Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Wikipedia subtraction rather than division) is 2^n where n is the number of digits. Is this correct? My experimentation* suggests that the number of subtractions required is > asymptotic to n^2. Mark * I'll post my code and its output if anyone wants but I'd prefer not to > clutter up Usenet with my abysmal C unnecessarily :-) The worst case is pretty bad. Just imagine finding the gcd of 2^{100} and 1 by subtraction. Your experimaentation could give interesting information about the average case. === Subject: Re: computational complexity of Euclid's algorithm The worst case is pretty bad. Just imagine finding the gcd of 2^{100} > and 1 by subtraction. Your experimaentation could give interesting > information about the average case. Yes, I was assuming that the Wikipedia link was referring to average case. I did try an analytic approach to the average case but didn't get very far. Mark === Subject: Re: computational complexity of Euclid's algorithm posting-account=VR0DOgoAAADggPTteFeA2AkmHNhjcrDV Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) The worst case is pretty bad. æJust imagine finding the gcd of 2^{100} > and 1 by subtraction. æYour experimaentation could give interesting > information about the average case. Yes, I was assuming that the Wikipedia link was referring to average > case. I did try an analytic approach to the average case but didn't get > very far. Mark There is a convention that unless one specifies otherwise, complexity means worst case. That makes sense, in a way, since a given problem may admit many reasonable notions of average case. But worst case may not be terribly relevant in the real world. For example, the worst case complexity of the Simplex Method is bad, but on average, and in practice, it behaves very well. The literature on the Euclidean Algorithm is huge. I am sure there is average case information available. Unfortunately I am too inexpert to provide references. === Subject: Re: computational complexity of Euclid's algorithm > There is a convention that unless one specifies otherwise, complexity > means worst case. That makes sense, in a way, since a given problem > may admit many reasonable notions of average case. > The literature on the Euclidean Algorithm is huge. I am sure there is > average case information available. Unfortunately I am too inexpert > to provide references. Mark === Subject: Re: computational complexity of Euclid's algorithm posting-account=lHNboAoAAACyasQ0uqX7OeM_tLuWGoQp 1.1.4322),gzip(gfe),gzip(gfe) > There is a convention that unless one specifies otherwise, complexity > means worst case. That makes sense, in a way, since a given problem > may admit many reasonable notions of average case. > The literature on the Euclidean Algorithm is huge. æI am sure there is > average case information available. æUnfortunately I am too inexpert > to provide references. Knuth Vol 2 gives an EXTENSIVE analyis. BTW, it is an old theorem that the worst two numbers are consecutive Fibonacci numbers (in terms of the work involved) === Subject: Re: 0.746081681128... ? > does anyone recognize this number ? 0.746081681128... ? Where does the reference come from? Do you know how it was computed? === Subject: Re: 0.746081681128... ? amy666 a .8ecrit : > does anyone recognize this number ? 0.746081681128... ? You may try this site: http://pi.lacim.uqam.ca/fra/ It answers this: http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookupf.pl?Submit=GO+&number=0.7460 81681128&lookup_type=browse -- Fatal === Subject: Re: 0.746081681128... ? posting-account=HGa-KwoAAACh0I68xblsVW5UyCAVbd5b .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > does anyone recognize this number ? > 0.746081681128... ? > tommy1729 Sure. Rounded to two decimal places, it's 3/4, the ratio of non-Olympics years in every quadrennial. === Subject: Re: 0.746081681128... ? amy666 a .8ecrit : > On Jul 20, 6:39 pm, amy666 > does anyone recognize this number ? > 0.746081681128... ? > tommy1729 >> Sure. Rounded to two decimal places, it's 3/4, >> the ratio of non-Olympics years in every >> quadrennial. your a real genius Calvin. Needs one to know one === Subject: Re: 0.746081681128... ? <7465888.1216672756261.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=HGa-KwoAAACh0I68xblsVW5UyCAVbd5b .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > On Jul 20, 6:39æpm, amy666 0.746081681128... ? > tommy1729 Sure. æRounded to two decimal places, it's 3/4, > the ratio of non-Olympics years in every > quadrennial. your a real genius Calvin. Sour your. === Subject: Re: Current public newsgroups posting-account=ee2apQoAAABJNMlLTFasCJw9Nfo9FmYk Gecko/20020924 AOL/7.0,gzip(gfe),gzip(gfe) HTTP/1.1 cache-ntc-aa10.proxy.aol.com[CFC8740E] (Traffic-Server/6.1.5 [uScM]) http://www.undergradresearch.psu.edu/displayByFaculty.cfm?fid=mwe1 More than 500 publications, why then you are still at the rank of > associate professor? > I have to say that current value judgments in the math community are pretty confused. In abstract math world we have reached to a state of research which results lost its purposes. If I had continued the same trend of research as I did for my qualitative research on ODE, certainly I would have been now very established mathematician at the rank of professor more than a decade ago (with more than 50 publications). But I happen to invest on NEW EXACT MATH, which will be with us for centuries to come I am pretty happy of my investment in this domain). This is a beautiful mathematics and views mathematical foundations from totally different perspective. Also this is a very different kind of math not practiced since a century ago. The established mathematicians mostly value a publication, if your research promoting their interest. I would say you have done abstract stuff in your number theory domain (at Penn State University) and invested in variety of publications in computer education domain (I am sure beneficial to the societyÍs interests). You should have been long years ago at professorship rank, but unfortunately the crisis in academia prevented their understanding of those who wanted to contribute in different aspects of education. What I wanted to say certainly it is not a disgrace to be left on associate professor level. But in your case I would say, it is misunderstanding of your university (Penn State University), not to value the educational contribution provided to the society at large by faculty members like you. We have a lot of issues to be resolved in the academia ranging from publications, institutions of research, positions of special groups, affirmative action issues in academia (http://mathforum.org/kb/thread.jspa?threadID=1773146&tstart=0), as well as funding research ,etc. The way China advanced its interests around the world (after entering WTO); I would say the USA Government should carefully look at its academic programs with China. Certainly issues like NSF collaboration of China [CapitalEth]USA should be dismantled and severe restrictions placed on chinies academia in the USA (along with possibly India, Poland and Russia). This will hopefully give a signal to China to carefully analyze its obligations in trade issues (which I strongly believe resulted in grave crisis in American economy). Otherwise we will see difficult problems remain intact in the society. I have decided that I eventually write my lecture notes on Maple files and disseminate it on CD (independent of those of math establishment). Basti Newsgroups I believe will eventually allow many others to be involved with newsgroups and express themselves in a healthy environment. Welcome to be a member at my newsgroups. Dr.M.Basti === Subject: Re: Current public newsgroups posting-account=ee2apQoAAABJNMlLTFasCJw9Nfo9FmYk Gecko/20020924 AOL/7.0,gzip(gfe),gzip(gfe) HTTP/1.1 cache-ntc-ab01.proxy.aol.com[CFC87441] (Traffic-Server/6.1.5 [uScM]) > I have to say that current value judgments in the math community are > pretty confused. In abstract math world we have reached to a state of research which > results lost its purposes. > The established mathematicians mostly value a publication, if your > research promoting their interest. > unfortunately the crisis in academia prevented their understanding of > those who wanted to contribute in different aspects of education. I have to emphasize those academic activities within educational issues of faculty members should be rewarded. This is essential to find a balance in strictly scholarly activities in areas of expertise and other activities beneficial to the public. Particularly in abstract math issues, many fields of research should be dismantled in the first place. Thus if you have published a few papers (50 of those more is the same and of no use). The establishment in math encourages research of faculty to enhance their results. In my particular case, I am leader of new fields of future math of NEW EXACT MATH, and it is not in my caliber to pursue the research of the establishment in abstract math. The prizes they arrange for themselves will not affect the crisis the math community is currently facing. In my case, the history repeated itself, and the math community behaved as examples of past and put a blind eye on this research for 2 decades. Why you think the history will not then repeat itself similar to other dark sides of the events in the 20th century? Dr.M.Basti === Subject: Re: Current public newsgroups posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Oh... I guess I must be one of those professors keeping you down and > keeping out the truth... Dr. Michael W. Ecker > Associate Professor of Mathematics > Pennsylvania State University > Wilkes-Barre Campus > Lehman, PA 18627 > please keep the good job you are doing. === Subject: Over-Feeding the babies, shanking the sperm doners. I know a guy who gets almost 900 USD per month ( for disablity ), but his drug habit ( crack, heroin ) has him ñ flying a sign î that reads: ñ Homeless vet î. He is a veteran, but not homeless .. his rent is 375 per month. Already, jobless mothers have it good .. I see it first-hand. They get free daycare, free college for themselves, checks ( cash ) in the mail, bad-ass apartments for a tiny fraction of those checks, free medical, free food ( lots of free food, WIC, food stamps, etc. ) free bus fair .. the list goes on and on. The ñ sperm donors î get the bill in the form of a nation-wide network of computers implementing automatic liens .. liens that Never Ever expire .. liens that no judge can touch. Interest and penalties of about 12 A.P.R. accrues on this debt. ( penalties vary by state ) How many planets do we have ? two ? no, just one, and these babies are burning it up like a log in the fire. Anything short of hibernation will shorten your life .. and even that could kill you toot sweet. Same goes for humanity as a whole. === Subject: Re: Over-Feeding the babies, shanking the sperm doners. posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > I know a guy who gets almost 900 USD per month ( for disablity ), > but his drug habit ( crack, heroin ) has him .81g flying a sign .81h > that reads: .81g Homeless vet .81h. He is a veteran, but not homeless .. his rent is 375 per month. Already, jobless mothers have it good .. I see it first-hand. > They get free daycare, free college for themselves, > checks ( cash ) in the mail, bad-ass apartments for a tiny fraction of those checks, > free medical, free food ( lots of free food, WIC, food stamps, etc. ) > free bus fair .. the list goes on and on. I admit I'm not an expert on it, but I don't doubt that single mothers now receive fairly good benefits. Actually my proposal is intended to economically benefit men more than women. > The .81g sperm donors .81h get the bill in the form of > a nation-wide network of computers implementing > automatic liens .. liens that Never Ever expire .. liens that no judge can touch. > Interest and penalties of about 12 A.P.R. accrues on this debt. > ( penalties vary by state ) If you're talking about mandatory child support, my proposal would allow us to eliminate it, or at least seriously reduce it. > How many planets do we have ? two ? no, just one, > and these babies are burning it up like a log in the fire. What? Andrew Usher === Subject: Re: Over-Feeding the babies, shanking the sperm donors. As I said: ñ Anything short of hibernation will shorten your life .. and even that could kill you toot sweet. Same goes for humanity as a whole. î. You don't understand that ? More ñ babies î spells more consumption, and, just as drinking too much ruins your health, overpopulation is ruining the health of the planet. As you guessed, I was taking about child suport when I said: ñ The ï sperm donors Í get the bill in the form of a nation-wide network of computers implementing automatic liens .. liens that Never Ever expire .. î. Society is over-feeding the babies and shanking the sperm donors. === Subject: Re: Help end wage-slavery posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) (changing the subject line back) > As I said: > .81g Anything short of hibernation will shorten your life .. > and even that could kill you toot sweet. > Same goes for humanity as a whole. .81h. You don't understand that ? More .81g babies .81h spells more consumption, > and, just as drinking too much ruins your health, > overpopulation is ruining the health of the planet. Yes, but you hardly expressed that. Your quote above appears to be meaningless. Andrew Usher === Subject: Generally, living faster means dying sooner. I'm sorry you feel the following is ñ meaningless î: ñ Anything short of hibernation will shorten your life .. and even that could kill you toot sweet. Same goes for humanity as a whole. î. It means that, generally, living faster means dying sooner. This is true for you, me, and humanity as a whole. === Subject: Re: Help end wage-slavery posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > I'm sorry you feel the following is .81g meaningless .81h: > .81g Anything short of hibernation will shorten your life .. > and even that could kill you toot sweet. > Same goes for humanity as a whole. .81h. It means that, generally, living faster means dying sooner. > This is true for you, me, and humanity as a whole. You're an idiot. And stop changing the subject line; that's NOT what it's for. Andrew Usher === Subject: Generally, living faster means dying sooner. Threads drift, the title should reflect that. ñ Except for houseflies, animal species tested with CR [ Caloric restriction ] so far, including primates, rats, mice, spiders, Drosophila, C. elegans and rotifers, have shown lifespan extension. CR is the only known dietary measure capable of extending maximum lifespan, as opposed to average lifespan. î. -- WikiPedia As I said: ñ Anything short of hibernation will shorten your life .. and even that could kill you toot sweet. Same goes for humanity as a whole. î. === Subject: QUIT CHANGING THE SUBJECT LINE, DUMBASS posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > Threads drift, the title should reflect that. === Subject: trendige herrenmode kaufen www policke herrenkleidung online de herrenbekleidung bestellen second herrenbekleidung bestellen outlet policke herrenkleidung online posting-account=nIHZkgoAAABDiYmT7mIHSsVRx4_m6iw4 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) trendige herrenmode kaufen www policke herrenkleidung online de herrenbekleidung bestellen second herrenbekleidung bestellen outlet policke herrenkleidung online + + + +++ MAENNERKLEIDUNG KAUFEN +++ MAENNERKLEIDUNG ONLINE BESTELLEN +++ + http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO http://WWW.MAENNERKLEIDUNG-KAUFEN-24.INFO + + + + + + + + +++ UEBERGROESSEN KAUFEN +++ UEBERGROESSEN ONLINE BESTELLEN +++ + http://WWW.UEBERGROESSEN-KAUFEN-24.INFO http://WWW.UEBERGROESSEN-KAUFEN-24.INFO http://WWW.UEBERGROESSEN-KAUFEN-24.INFO http://WWW.UEBERGROESSEN-KAUFEN-24.INFO http://WWW.UEBERGROESSEN-KAUFEN-24.INFO http://WWW.UEBERGROESSEN-KAUFEN-24.INFO + + + indische herrenmode kaufen herrenbekleidung bestellen bad elegante herrenmode kaufen herrenmode kaufen onlineshop windsor herrenbekleidung bestellen elegante herrenmode kaufen gutmann herrenbekleidung bestellen === Subject: Cancel Re: A consideration concerning the diagonal argument of G. Cantor(was Q) === Subject: more experiments with products of sines In January 2007, Leo Wapner asked about the convergence or divergence of 2|sin(1)| 2|sin(2)| 2|sin(3)| ... In the message I did some computations related to a convergent to pi. rational multiple radian approximation of pi: { unit = pi/22} 1 19 2 16 3 13 4 10 5 7 6 4 7 1 8 20 9 17 10 14 11 11 12 18 13 5 14 2 15 21 16 18 17 15 18 12 19 9 20 6 21 3 How to produce this: 7*pi/22 ~= 1, so 14*pi/22 ~=2, 21*pi/22~=3, 28*pi/22 ==6*pi/22 (mod pi), so 6*pi/22 ~= 4 rad (mod pi), and so on. The product | sin(pi/22) sin(2*pi/22) .... sin(21*pi/22) | = 22/(2^21) is known, cf. Robert Israel's message following for more details. Also, in 1982 in a survey of developments in hyperbolic geometry, John Milnor gave a sketch of a proof of a slightly more general result: Cf For x in ]0, pi[, d/dx log(sin(x)) = cotan(x). The following is a table produced by a computer program, comparing |sin(k*pi/22)| to |sin( n(k) )|, where n is the corresponding integer in the radian approximation column above. More precisely, the column dif. has log( |sin( n(k) )| ) - log( |sin(k*pi/22)| ) . By taking the log of |sin(1)| |sin(2)| |sin(3)| ... |sin(21)| and subtracting log(Prod_{k=1 ... 21} | sin(k*pi/22)| , we get the sum of the dif. values. The der. (for derivative) is the largest of |cotan(k*pi/22)| and |cotan ( n(k) ) |, to be on the safe side, using the MVT. The column dx is the absolute value of the difference between k*pi/22 and n(k) (mod pi). The column apdif is defined by: apdif : = der. * dx [ epsilon = 0.00040234 := 1 - 7*pi/22 ] (A) rational multiple (B) radian approximation of pi: { unit = pi/22} (A) (B) (dif. of logs) max. of deriv. delta_x | approx. diff. of logs| 1 19 dif.=0.051775 der.=6.9552 dx=0.007644 apdif=0.0532 2 16 dif.=0.021666 der.=3.4057 dx=0.006437 apdif=0.0219 3 13 dif.=0.011374 der.=2.1897 dx=0.005230 apdif=0.0115 4 10 dif.=0.006233 der.=1.5560 dx=0.004023 apdif=0.0063 5 7 dif.=0.003241 der.=1.1541 dx=0.002816 apdif=0.0033 6 4 dif.=0.001392 der.=0.8665 dx=0.001609 apdif=0.0014 7 1 dif.=0.000258 der.=0.6427 dx=0.000402 apdif=0.0003 8 20 dif.=0.003636 der.=0.4567 dx=0.008047 apdif=0.0037 9 17 dif.=0.001983 der.=0.2936 dx=0.006840 apdif=0.0020 10 14 dif.=0.000794 der.=0.1438 dx=0.005633 apdif=0.0008 11 11 dif.=-0.000010 der.=0.0044 dx=0.004426 apdif=0.0000 12 8 dif.=-0.000468 der.=0.1471 dx=0.003219 apdif=0.0005 13 5 dif.=-0.000593 der.=0.2958 dx=0.002012 apdif=0.0006 14 2 dif.=-0.000368 der.=0.4577 dx=0.000805 apdif=0.0004 15 21 dif.=-0.005481 der.=0.6547 dx=0.008449 apdif=0.0055 16 18 dif.=-0.006321 der.=0.8793 dx=0.007242 apdif=0.0064 17 15 dif.=-0.007007 der.=1.1682 dx=0.006035 apdif=0.0071 18 12 dif.=-0.007553 der.=1.5727 dx=0.004828 apdif=0.0076 19 9 dif.=-0.007967 der.=2.2108 dx=0.003621 apdif=0.0080 20 6 dif.=-0.008258 der.=3.4364 dx=0.002414 apdif=0.0083 21 3 dif.=-0.008431 der.=7.0153 dx=0.001207 apdif=0.0085 The largest dif. values (in absolute value) are for pi/22, pi/22 and 3*pi/22. For those values, the sine function is small and the derivative of log(sin(x)), or cotan(x), is quite large; also, the quantities dx, proportional to n(k), are large. So dif. is large, and apdif is quite close. By comparison, log(22) for the log of the known product is about 3.09 . For n(k) = 1, delta_x = | 7*pi/22 - 1| = 7/22 * | pi - 22/7| and here we could use the bound in Hurwitz' theorem for the general case. The other thing of interest is getting an upper bound on the average of the 21 absolute values of cotan used together with the delta_x's and the MVT to get the apdif's. Any delta_x is < 22* 7/22 * | pi - 22/7| = 7* |pi - 22/7| . I will probably try with other convergents to pi. The C source code is copied below... David Bernier #include #include int main(void) { long n, k; double mypi = 3.14159265358979323846; double diff; double epsilon; double approx; double delta_x; double derivative, derivative2; epsilon = mypi * ((double)7)/((double)22); epsilon = fabs(epsilon - ((double)1) ); printf(epsilon = %0.8lfnn, epsilon); for(k=1;k<22;k++) { n = (19*k)%22; diff = log(fabs(sin( (double)n ))); diff = diff - log(fabs(sin( mypi * ((double)k)/((double)22)))); derivative = ((double)1)/tan(mypi * ((double)k)/((double)22)); derivative = fabs(derivative); derivative2 = ((double)1)/tan((double)n); derivative2 = fabs(derivative2); if(derivative2>derivative) { derivative = derivative2; } delta_x = ((double)n)*epsilon; delta_x = fabs(delta_x); approx = delta_x*derivative; printf(%ldt%ldtdif.=%.6lftder.=%.4lftdx=%.6lftapdif=%.4lfn, k, n, diff,derivative,delta_x,approx); } printf(nn); return 0; } === Subject: EC-funded Research Visits: Application Deadline 31/08/2008 HPC-EUROPA++ deadline for applications: 31 August 2008 ---------------------------------------------------- It's a very fruitful experience. You can share your knowledge, discover new field of research, see a different way of working. HPC-Europa++ Visitor, 2008 - Are you a scientist of postgraduate level or above, working in EU or an Associated State? - Do you require large computing power to improve your research? - Would you like to visit a similar research group in Italy, Spain, UK, The Netherlands, France, or Germany? ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ EC-funded HPC-Europa++ Visitor programme is the answer ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ See what we can offer you: - Access to the best high performance computing systems in Europe - Technical support and consultancy - Scientific collaboration with a host researcher - Travel and living expenses - Logistical and administrative support - Minimal paperwork See http://www.hpc-europa.org/ta.html for more information and online application form. 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Alain === Subject: Recommendations for effective online scientific discussion This set of recommendations was first prepared after noticing some nasty behaviors on the USENET. Initially, I embraced the usual recommendation to ignore trolls, crackpots, liars, flamers, and other nasty posters. This worked for most of cases. For example, after postings from certain nasty posters were ignored several times, the posters stopped from replying more. However, this recommendation did not work for the most nasty individuals [*] and this has obligated me to rethink the guidelines, incorporating new tactics for beating that people. I had been ignoring most accusations, nonsenses, and lies from certain poster in non-moderated newsgroups but then he followed me to two moderated newsgroup and started again. Also there was some recent discussion in moderated newsgroup sci.physics.foundations about moderation policies, just after a group of people openly protested to the moderator board about the annoying messages that passed moderation. This chapter finished after one of the moderators warned in public to the nasty poster. Since his interest in that moderated newsgroup has diminished and actually he is not posting more, returning to moderated newsgroups where he posts dozens of nasty messages (insults, lies, ad hominem...) each day. After this episode was analyzed I have also included recommendations for moderated newsgroups. I have done several experiments in different newsgroups, forcing limiting behaviors, recollecting the data, and preparing the guidelines. In several occasions I have publicly said I was recollecting information for the guidelines and warned nasty posters about their behavior. For > I repeat again, are you aware that this thread and your messages will be > cited in a new version of USENET guidelines? And the reply was Why should I care? Nobody but you cares about your guidelines. I have also received some feedback from users of moderated newsgroups, I wait more feedback from moderators for this part. Another defect of the original recommendation to ignore crackpots was they feel freedom to post anything nonsensical they want. This was irrelevant for veteran readers, of course, most of whom already kill- filled the crackpots but was a source of continuous confusion for novice readers. relativity and one crackpot replied No. The special relativistic Hamiltonian is H = L = -mc^2 * [1 - v^2/ c^2 ]. and from here followed a nasty discussion with more crackpots adding more mistakes, noise, and insults. For example, another crackpot picked over http://sci.tech-archive.net/Archive/sci.physics.relativity/2008-07/ msg00824.html Everyone who studied a minimum of special relativity and mechanics detected the mistakes they were doing, but there was case of one novice (no physicists are usual on newsgroups) who did not and supported in I think it is our responsibility do not ignore those notorious mistakes and warn novice readers always was possible. That is my current view with further recommendations to notice the mistakes without being caught in the noise generated by crackpots as those. The guidelines are accessible on http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html their reading of the guidelines and useful suggestions. NOTE [*] There is a documented case of a clearly perturbed individual who has tried for close a year to falsify the Groups ratings about posters adding hundred of negative stars and simulating the ratings of a hundred of different readers. He achieved this by exploiting a well-known bug in the Groups console. This poster was once caught by a Groups administrator, and his fake rates eliminated from the total rate. But that did not stop him. He changed the nick, started a new account and continues trying to falsify the archive. This and other samples from crackpots like Eric Gisse, disorder trolls as Dono (Karandash2), professional liars as Tom Roberts and others will be presented, with details, dates, original messages, and links on a future Micro-thought, Some samples of USENET fauna, now in preparation canonicalscience.blogspot.com -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === Subject: Re: Recommendations for effective online scientific discussion posting-account=7bF0GwoAAABMFHX6V4fON4-1F6LFJ834 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) Restored the full cross-post list. On Jul 22, 2:22æam, Juan R. Gonz.87lez-.8dlvarez > This set of recommendations was first prepared after > noticing some nasty behaviors on the USENET. ... > I think it is our responsibility do not ignore those > notorious mistakes and warn novice readers always > was possible. That is my current view with further > recommendations to notice the mistakes without > being caught in the noise generated by crackpots > as those. Sure had to cut out a lot of whining, trolling, and personal opinion before I got to your recommendation. Why do you trim the Followup-To list on a posting such as this? If in all five groups. David A. Smith === Subject: Re: Recommendations for effective online scientific discussion > This set of recommendations was first prepared after noticing some nasty > behaviors on the USENET. You might want to start, Juan, by correcting your own bad Usenet behavior. Do not abuse the followup-to header. Why did you think your post needed to appear on five groups, but replies should go only to sci.physics.relativity? [...] > In several occasions I have publicly said I was recollecting information > for the guidelines and warned nasty posters about their behavior. For > I repeat again, are you aware that this thread and your messages will be >> cited in a new version of USENET guidelines? And the reply was Why should I care? Nobody but you cares about your guidelines. That strikes me as a reasonable response. [...] > This and other samples from crackpots like Eric Gisse, disorder trolls as > Dono (Karandash2), professional liars as Tom Roberts and others will be > presented, with details, dates, original messages, and links on a future Who do you think you are kidding, Juan? What distinguishes these people isn't outstandingly nasty Usenet behavior; it's that they argued with *you*. Now you have the nerve to pretend your issue is effective online scientific discussion. No one is fooled. Juan, your real cause could not be more obvious. -- --Bryan === Subject: Re: Recommendations for effective online scientific discussion pan.2008.07.22.09.23.57@canonicalscience.com > This and other samples from crackpots like Eric Gisse, disorder trolls as > Dono (Karandash2), professional liars as Tom Roberts ... It follows guidelines: http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/IFollowGuidelines.htm l http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/Guidelines.html Dirk Vdm === Subject: Re: Recommendations for effective online scientific discussion <9vnhk.2406$5O6.2363@newsfe14.ams2> posting-account=nf79RwoAAABXjvy5ztMzmPxgY1WGoktI Gecko/20080201 Firefox/2.0.0.12,gzip(gfe),gzip(gfe) > pan.2008.07.22.09.23...@canonicalscience.com > This and other samples from crackpots like Eric Gisse, disorder trolls as > Dono (Karandash2), professional liars as Tom Roberts ... > It follows guidelines: > http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/IFollowGuideli... > http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/Guidelines.html Dirk Vdm Proof positive that this topic author Juan R. Gonz.87lez-.8dlvarez is alt.creep bigot you actually are. - Brad Guth Brad Guth Brad.Guth BradGuth === Subject: Re: Recommendations for effective online scientific discussion posting-account=nf79RwoAAABXjvy5ztMzmPxgY1WGoktI Gecko/20080201 Firefox/2.0.0.12,gzip(gfe),gzip(gfe) On Jul 22, 2:22 am, Juan R. Gonz.87lez-.8dlvarez > This set of recommendations was first prepared after noticing some nasty > behaviors on the USENET. Initially, I embraced the usual recommendation > to ignore trolls, crackpots, liars, flamers, and other nasty posters. This worked for most of cases. For example, after postings from certain > nasty posters were ignored several times, the posters stopped from > replying more. However, this recommendation did not work for the most nasty individuals > [*] and this has obligated me to rethink the guidelines, incorporating > new tactics for beating that people. I had been ignoring most accusations, nonsenses, and lies from certain > poster in non-moderated newsgroups but then he followed me to two > moderated newsgroup and started again. Also there was some recent discussion in moderated newsgroup > sci.physics.foundations about moderation policies, just after a group of > people openly protested to the moderator board about the annoying > messages that passed moderation. This chapter finished after one of the > moderators warned in public to the nasty poster. Since his interest in that moderated newsgroup has diminished and > actually he is not posting more, returning to moderated newsgroups where > he posts dozens of nasty messages (insults, lies, ad hominem...) each day. After this episode was analyzed I have also included recommendations for > moderated newsgroups. I have done several experiments in different > newsgroups, forcing limiting behaviors, recollecting the data, and > preparing the guidelines. In several occasions I have publicly said I was recollecting information > for the guidelines and warned nasty posters about their behavior. For I repeat again, are you aware that this thread and your messages will be > cited in a new version of USENET guidelines? And the reply was Why should I care? Nobody but you cares about your guidelines. I have also received some feedback from users of moderated newsgroups, I > wait more feedback from moderators for this part. Another defect of the original recommendation to ignore crackpots was > they feel freedom to post anything nonsensical they want. This was > irrelevant for veteran readers, of course, most of whom already kill- > filled the crackpots but was a source of continuous confusion for novice > readers. relativity and one crackpot replied No. The special relativistic Hamiltonian is H = L = -mc^2 * [1 - v^2/ > c^2 ]. and from here followed a nasty discussion with more crackpots adding more > mistakes, noise, and insults. For example, another crackpot picked over http://sci.tech-archive.net/Archive/sci.physics.relativity/2008-07/ > msg00824.html Everyone who studied a minimum of special relativity and mechanics > detected the mistakes they were doing, but there was case of one novice > (no physicists are usual on newsgroups) who did not and supported in I think it is our responsibility do not ignore those notorious mistakes > and warn novice readers always was possible. That is my current view with > further recommendations to notice the mistakes without being caught in > the noise generated by crackpots as those. The guidelines are accessible on http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html their reading of the guidelines and useful suggestions. NOTE [*] There is a documented case of a clearly perturbed individual who has > tried for close a year to falsify the Groups ratings about posters adding > hundred of negative stars and simulating the ratings of a hundred of > different readers. He achieved this by exploiting a well-known bug in the > Groups console. This poster was once caught by a Groups administrator, and his fake rates > eliminated from the total rate. But that did not stop him. He changed the > nick, started a new account and continues trying to falsify the archive. This and other samples from crackpots like Eric Gisse, disorder trolls as > Dono (Karandash2), professional liars as Tom Roberts and others will be > presented, with details, dates, original messages, and links on a future > Micro-thought, Some samples of USENET fauna, now in preparation canonicalscience.blogspot.com -- > Center for CANONICAL |SCIENCE) http://canonicalscience.org How or why the hell does ñJuan R. Gonz.87lez-.8dlvarezî manage to crosspost a given topic into sci.physics whereas replies to this simply never materialize as contributed to any of this topic as having been crossposted into this otherwise public newsgroup? It functions as though the topic is broken, even though all other sci.physics topics are working properly. It must be because of using whole lot of sense if ñJuan R. Gonz.87lez-.8dlvarezî is trying to draw public attention to his/her rant. Recommendations for effective online scientific discussion option? - Brad Guth Brad Guth Brad.Guth BradGuth === Subject: =?windows-1252?Q?Munifus_Landmark_Estate_=99?= posting-account=uWkSGAoAAABcv1Qo8K_q913gt5Szmp28 SV1),gzip(gfe),gzip(gfe) Home Manufacturing Worldwide Munifus Landmark Estate » represents a building-contractor located in many of the most beautiful and highly attractive locations around the world. The following represents a detailed description for one selection of Munifus Luxury Property ». 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Reservation: Way/Road Information: Prize Indices Analysis, Insurance, Inspection, Finance. Agent: Mr Roger K. Olsson Int call: +46 (0) 705474830 Prospect: http://publishing.yudu.com/Freedom/Ak8i8/DevelopmentProject/ Enquiry Mail to: Munifus Landmark Estate » Mr Roger K. Olsson PL 2540 Kuttainen SE: 98016 Karesuando === Subject: Novice - commutator subgroups posting-account=NoOusQoAAADeMejejmF9SPjc_TwhqzV0 SLCC1; .NET CLR 2.0.50727; .NET CLR 3.0.04506; Media Center PC 5.0; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Let H be a subgroup of a finite group G and let s be an element of G. If [H,] = H', is it true that s commutes with each element in H-H' ? === Subject: Re: Novice - commutator subgroups > Let H be a subgroup of a finite group G and let s be > an element of G. If [H,] = H', is it true that s > commutes with each element in H-H' ? No. Try an example. You may also consider how much more interesting your questions might be with context and followup. You asked another question about 2-groups which was false for obvious examples. Why do you think these things are true? === Subject: Re: Novice - commutator subgroups >> Let H be a subgroup of a finite group G and let s be >> an element of G. If [H,] = H', is it true that s >> commutes with each element in H-H' ? No. Try an example. You may also consider how much more interesting your > questions might be with context and followup. You > asked another question about 2-groups which was > false for obvious examples. Why do you think these > things are true? Since I noticed that basically every non-trivial group was a counterexample, I thought it might be interesting to find out exactly which groups work. For G a finite group and s in G, call the pair (G,s): * good if [G,G] = [G,] and s commutes with G-[G,G] * bad if [G,G] = [G,] but s does not commute with G-[G,G] Call a finite group G: * good if it has a good pair. * great if it has a good pair, but has no bad pair. Here is are some easy exercises: * If G is good, then G is either abelian or perfect. * If G is abelian or perfect, then G is great. Most of this is very easy, and is just the statement that a group is not the union of two proper subgroups. Showing perfect groups are great is a little longer, and struck me as requiring the fact that a finite perfect group is not the union of its proper normal subgroups. It is clear a perfect group can have no bad pair, since commuting with G-[G,G] is vacuous. However, showing the existence of an s such that G=[G,] might be a little harder, and is the only place I used any finiteness condition. Here I used that a finite perfect group G is not the union of its proper normal subgroups. This is true since it only has finitely many normal subgroups, or more precisely, every proper normal subgroup is contained in a maximal normal subgroup, and there are only finitely many maximal normal subgroups. Then the quotient of G by the intersection of its maximal normal subgroups is a finite direct product of nonabelian simple groups. The normal subgroups of such a group are well known, and any element of the group with nonidentity projections onto each direct factor is an s to create a good pair. === Subject: Integral with sine and square root posting-account=zzlw9goAAAAa9-yEZ5p3rH0AyeiHu4Cp Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hello everyone. I try to resolve the following integral: int sin(sqrt(x^2+a^2)-phi) over x. The parameters a and phi are constant and known. Do you have any idea about how to do this? Dali === Subject: Re: Integral with sine and square root Well said sir === Subject: Re: Integral with sine and square root Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) http://integrals.wolfram.com/index.jsp?expr=sin%28sqrt%28x%5E2%2B1%29%29&ran dom=false didn't seem to help . === Subject: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation in order to compute time dilation for a clock-carrying traveler? Shubee http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? OK bert, This has been going on too long. There is no time dilation for the clock carrying traveller. So you are asking a stupid question that reflects your level of (mis) understanding of SR. There is time dilation for any DIFFERENT observer IN MOTIOn wrt the traveller that is carrying the clock. As you have been told by several others, the time dilation can be calculated directly by the sole application of the Lorentz tranforms: t=gamma(tau-zeta*v/c^2) t=coordinate time tau=proper time (on the traveller clock) zeta=proper distance So: For dzeta=0 dt=gamma*d(tau) dt= === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=V47zDAoAAAANjA0PBqp-kz4RtUTJrqYB Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf Edward Green said it best: None Ben Green (no relation) === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf Edward Green said it best: None Ben Green (no relation) OK U Ben, Please answer the same question I asked Edward Green: So how would you prove that the Lorentz transformation implies time dilation without presupposing more than just the Lorentz transformation? Shubee http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=V47zDAoAAAANjA0PBqp-kz4RtUTJrqYB Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf Edward Green said it best: None Ben Green (no relation) OK U Ben, Please answer the same question I asked Edward Green: So how would you prove that the Lorentz transformation implies time > dilation without presupposing more than just the Lorentz > transformation? Shubeehttp://www.everythingimportant.org/relativity/special.pdf All you have to do is to construct an example. Imagine two points on the x axis of an inertial frame. Imagine an event1 at the origin at time t=0 and an event2 at x=1 at time t=5. Then imagine a traveller travelling w.r.t. S at constant velocity 0.1 c (w.r.t. S) along the x axis. Associate another inertial frame with the traveller in which he is at rest. Call his frame S'. Suppose event 1 w.r.t. S' is at x'=0, t'=0 without loss of generality. Then use the Lorentz Transformation to find the coordinates of event2 w.r.t. S' by calculating x and t of event2. You should find that event 2 has t' less than t. That would be a QED. Clear? Uncle Ben === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=V47zDAoAAAANjA0PBqp-kz4RtUTJrqYB Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf Edward Green said it best: None Ben Green (no relation) OK U Ben, Please answer the same question I asked Edward Green: So how would you prove that the Lorentz transformation implies time > dilation without presupposing more than just the Lorentz > transformation? Shubeehttp://www.everythingimportant.org/relativity/special.pdf All you have to do is to construct an example. Imagine two points on > the x axis of an inertial frame. Imagine an event1 at the origin at > time t=0 and an event2 at x=1 at time t=5. Then imagine a traveller travelling w.r.t. S at constant velocity 0.1 > c (w.r.t. S) along the x axis. Associate another inertial frame with > the traveller in which he is at rest. Call his frame S'. Suppose event 1 w.r.t. S' is at x'=0, t'=0 without > loss of generality. Then use the Lorentz Transformation to find the > coordinates of event2 w.r.t. S' > by calculating x and t of event2. You should find that event 2 has t' less than t. That would be a QED. Clear? Uncle Ben The double primes should be single primes. Sorry. === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf Edward Green said it best: None Ben Green (no relation) OK U Ben, Please answer the same question I asked Edward Green: So how would you prove that the Lorentz transformation implies time > dilation without presupposing more than just the Lorentz > transformation? Calculate how fast a clock ticks using the Lorentz transformation. Why is this a mystery? Shubeehttp://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? >> What assumptions, if any, must be added to the Lorentz transformation >> in order to compute time dilation for a clock-carrying traveler? >> Shubeehttp://www.everythingimportant.org/relativity/special.pdf > Edward Green said it best: None > Ben Green (no relation) >> OK U Ben, >> Please answer the same question I asked Edward Green: >> So how would you prove that the Lorentz transformation implies time >> dilation without presupposing more than just the Lorentz >> transformation? Calculate how fast a clock ticks using the Lorentz transformation. Why is this a mystery? Sometimes one wonders whether he is just pretending to be such an infinite dope - just to discredit the other dopes around here. Amazing. Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Sometimes one wonders whether he is just pretending to be > such an infinite dope - just to discredit the other dopes > around here. > Amazing. Dirk Vdm For my next trivial yet impossible to believe theorem at http://www.everythingimportant.org/relativity/special.pdf I shall start with the Lorentz transformation and the clock synchronization scheme it implies and then reset all clocks in all but one inertial distance in all but that one special inertial frame by a suitable scaling factor. I shall also fiddle with clock rates by applying a similar suitable change of time scale and thus finally arrive at the traditional and easy to understand Galilean group of transformations. This will prove that the Lorentz transformation alone doesn't imply time dilation. Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=V47zDAoAAAANjA0PBqp-kz4RtUTJrqYB Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Sometimes one wonders whether he is just pretending to be > such an infinite dope - just to discredit the other dopes > around here. > Amazing. Dirk Vdm For my next trivial yet impossible to believe theorem athttp://www.everythingimportant.org/relativity/special.pdfI shall > start with the Lorentz transformation and the clock synchronization > scheme it implies and then reset all clocks in all but one inertial > distance in all but that one special inertial frame by a suitable > scaling factor. I shall also fiddle with clock rates by applying a > similar suitable change of time scale and thus finally arrive at the > traditional and easy to understand Galilean group of transformations. This will prove that the Lorentz transformation alone doesn't imply > time dilation. Shubee Yes, but is it allowed to do all this fiddling? (Written after posting a worked example.) Uncle Ben === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Sometimes one wonders whether he is just pretending to be > such an infinite dope - just to discredit the other dopes > around here. > Amazing. Dirk Vdm For my next trivial yet impossible to believe theorem at > http://www.everythingimportant.org/relativity/special.pdf I shall > start with the Lorentz transformation and the clock synchronization > scheme it implies and then reset all clocks in all but one inertial > distance in all but that one special inertial frame by a suitable > scaling factor. I shall also fiddle with clock rates by applying a > similar suitable change of time scale and thus finally arrive at the > traditional and easy to understand Galilean group of transformations. This will prove that the Lorentz transformation alone doesn't imply > time dilation. Shubee Yes, but is it allowed to do all this fiddling? UB, My derivation of the Lorentz transformation is from the weakest axiom set possible. For example, my approach doesn't presuppose any significant conditions on the comparative distance and time scales for different frames of reference. So why can't I invoke an arbitrary change of scale afterwards? http://www.everythingimportant.org/relativity/special.pdf Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? >> Sometimes one wonders whether he is just pretending to be >> such an infinite dope - just to discredit the other dopes >> around here. >> Amazing. >> Dirk Vdm > For my next trivial yet impossible to believe theorem at > http://www.everythingimportant.org/relativity/special.pdf I shall > start with the Lorentz transformation and the clock synchronization > scheme it implies and then reset all clocks in all but one inertial > distance in all but that one special inertial frame by a suitable > scaling factor. I shall also fiddle with clock rates by applying a > similar suitable change of time scale and thus finally arrive at the > traditional and easy to understand Galilean group of transformations. > This will prove that the Lorentz transformation alone doesn't imply > time dilation. > Shubee >> Yes, but is it allowed to do all this fiddling? UB, My derivation of the Lorentz transformation is from the weakest ... part of your inferior brain? Yes, we know. Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Sometimes one wonders whether he is just pretending to be > such an infinite dope - just to discredit the other dopes > around here. > Amazing. Dirk Vdm For my next trivial yet impossible to believe theorem athttp://www.everythingimportant.org/relativity/special.pdfI shall > start with the Lorentz transformation and the clock synchronization > scheme it implies and then reset all clocks in all but one inertial > distance in all but that one special inertial frame by a suitable > scaling factor. I shall also fiddle with clock rates by applying a > similar suitable change of time scale and thus finally arrive at the > traditional and easy to understand Galilean group of transformations. This will prove that the Lorentz transformation alone doesn't imply > time dilation. Shubee Sorry Shooby, modern special relativity is not defined by either the Lorentz transformation or clock synchronization. In fact, no mention of light is made at all. You know this of course, but that prevents you from playing semantic games that purposefully confuse the issue. === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Sometimes one wonders whether he is just pretending to be > such an infinite dope - just to discredit the other dopes > around here. > Amazing. Dirk Vdm For my next trivial yet impossible to believe theorem athttp://www.everythingimportant.org/relativity/special.pdfI shall > start with the Lorentz transformation and the clock synchronization > scheme it implies and then reset all clocks in all but one inertial > distance in all but that one special inertial frame by a suitable > scaling factor. I shall also fiddle with clock rates by applying a > similar suitable change of time scale and thus finally arrive at the > traditional and easy to understand Galilean group of transformations. This will prove that the Lorentz transformation alone doesn't imply > time dilation. Shubee Doesn't anyone remember me calling this result the most entertaining exploit in the history of physics? Shubee http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > Calculate how fast a clock ticks using the Lorentz transformation. Why is this a mystery? It is not a mystery at all, http://www.hyperdeath.co.uk/spaceman/message.html :) -- James M Driscoll Jr Spaceman === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? | What assumptions, if any, must be added to the Lorentz transformation | in order to compute time dilation for a clock-carrying traveler? Deletion must be added, time dilation doesn't happen. Why did Einstein say the speed of light from A to B is c-v, the speed of light from B to A is c+v, the time each way is the same? Easy: he did NOT say that. - cretin harald.vanlintelButNotThis@epfl.ch According to xxein: It is an artefactual/superficially imposed yin-yang of sorts. According to Lamenting Shubert: Why do you want to know? In neither system (meaning frame of reference in modern-day terminology) is the speed of light c-v or c+v. In both systems the speed of light is c. -- cretin Jimmy Black fmlast3@organization.edu. According to the imbecile Jimmy Black, Einstein did not write the equation === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? This wording (time dilation *for* a clock-carrying traveler?) reveals a definitely sad amount of ignorance. Shubee > http://www.everythingimportant.org/relativity/special.pdf Relevant equation: Dt' = gamma ( Dt - v Dx ) Assumption: A clock is at rest in the unprimed frame, i.o.w. Dx = 0 Result: Dt' = gamma Dt whence Dt' > Dt meaning that in the unprimed frame the time between two ticks of the (moving) clock is dilated with respect to the time between those ticks as measured in the rest frame of the clock. General conclusion: Physics is not merely an application of algebra. Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? This wording (time dilation *for* a clock-carrying traveler?) > reveals a definitely sad amount of ignorance. Shubee >http://www.everythingimportant.org/relativity/special.pdf Relevant equation: > Dt' = gamma ( Dt - v Dx ) Assumption: > A clock is at rest in the unprimed frame, i.o.w. > Dx = 0 Result: > Dt' = gamma Dt > whence > Dt' > Dt > meaning that in the unprimed frame the time between two ticks of > the (moving) clock is dilated with respect to the time between those > ticks as measured in the rest frame of the clock. General conclusion: > Physics is not merely an application of algebra. Dirk Vdm This is what you get for trying to teach bert :-) === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? >> This wording (time dilation *for* a clock-carrying traveler?) >> reveals a definitely sad amount of ignorance. > Shubee > http://www.everythingimportant.org/relativity/special.pdf >> Relevant equation: >> Dt' = gamma ( Dt - v Dx ) >> Assumption: >> A clock is at rest in the unprimed frame, i.o.w. >> Dx = 0 >> Result: >> Dt' = gamma Dt >> whence >> Dt' > Dt >> meaning that in the unprimed frame the time between two ticks of >> the (moving) clock is dilated with respect to the time between those >> ticks as measured in the rest frame of the clock. >> General conclusion: >> Physics is not merely an application of algebra. >> Dirk Vdm This is what you get for trying to teach bert :-) HUH. Teach bert? Utterly impossible. Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? > This wording (time dilation *for* a clock-carrying traveler?) >> reveals a definitely sad amount of ignorance. >> Shubee >http://www.everythingimportant.org/relativity/special.pdf > Relevant equation: >> Dt' = gamma ( Dt - v Dx ) > Assumption: >> A clock is at rest in the unprimed frame, i.o.w. >> Dx = 0 > Result: >> Dt' = gamma Dt >> whence >> Dt' > Dt >> meaning that in the unprimed frame the time between two ticks of >> the (moving) clock is dilated with respect to the time between those >> ticks as measured in the rest frame of the clock. > General conclusion: >> Physics is not merely an application of algebra. > Dirk Vdm This is what you get for trying to teach bert :-) HUH. > Teach bert? > Utterly impossible. Dirk Vdm You absolutely need to add this pearl, Juano correcting Eric: Pay attention to: The third mistake being his confusion between the Lagrangian and the energy in special relativity. Note that Eric *multiplies* by the factor (1 - (v^2/ c^2)) instead DIVIDING by it. The master at his best :-) === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? >> This wording (time dilation *for* a clock-carrying traveler?) >> reveals a definitely sad amount of ignorance. > Shubee > http://www.everythingimportant.org/relativity/special.pdf >> Relevant equation: >> Dt' = gamma ( Dt - v Dx ) >> Assumption: >> A clock is at rest in the unprimed frame, i.o.w. >> Dx = 0 >> Result: >> Dt' = gamma Dt >> whence >> Dt' > Dt >> meaning that in the unprimed frame the time between two ticks of >> the (moving) clock is dilated with respect to the time between those >> ticks as measured in the rest frame of the clock. >> General conclusion: >> Physics is not merely an application of algebra. >> Dirk Vdm > This is what you get for trying to teach bert :-) >> HUH. >> Teach bert? >> Utterly impossible. >> Dirk Vdm You absolutely need to add this pearl, Juano correcting Eric: > Pay attention to: The third mistake being his confusion between the Lagrangian and the > energy in special relativity. Note that Eric *multiplies* by the > factor (1 - (v^2/ c^2)) instead DIVIDING by it. The master at his best :-) Yes, I had seen this, but alas, it is insufficiently self-explaining to serve as an entry. And besides, he should have multiplied with the square root of that factor :-) Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > Yes, I had seen this, but alas, it is insufficiently self-explaining to > serve as an entry. > And besides, he should have multiplied with the square root > of that factor :-) Dirk Vdm- He likes to be reminded on a daily basis :-) === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? This wording (time dilation *for* a clock-carrying traveler?) > reveals a definitely sad amount of ignorance. I noticed that you repeated the same definitely sad amount of ignorance below. > Shubee > http://www.everythingimportant.org/relativity/special.pdf Relevant equation: > Dt' = gamma ( Dt - v Dx ) Assumption: > A clock is at rest in the unprimed frame, i.o.w. > Dx = 0 Result: > Dt' = gamma Dt > whence > Dt' > Dt > meaning that in the unprimed frame the time between two ticks of > the (moving) clock is dilated with respect to the time between those > ticks as measured in the rest frame of the clock. General conclusion: > Physics is not merely an application of algebra. Correct, which means that we should ignore your ignorant application of algebra and complete misunderstanding of what the Lorentz transformation implies. Shubee http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation [...] >> Relevant equation: >> Dt' = gamma ( Dt - v Dx ) >> Assumption: >> A clock is at rest in the unprimed frame, i.o.w. >> Dx = 0 >> Result: >> Dt' = gamma Dt >> whence >> Dt' > Dt >> meaning that in the unprimed frame the time between two ticks of >> the (moving) clock is dilated with respect to the time between those >> ticks as measured in the rest frame of the clock. >> General conclusion: >> Physics is not merely an application of algebra. Correct, which means that we should ignore your ignorant application > of algebra and complete misunderstanding of what the Lorentz > transformation implies. Shubee, did you try it derive it yourself? I usually need to start with a specific example, and work toward something more general. Start with two coordinate systems in standard configuration. In the one defined as at rest, put a clock at x=0. Each tick of the clock marks a second in the rest frame, and the Lorentz transform will map the tick events to a time (and place, but that's not the question here) in the moving frame, which tells us how fast the clock is running in the moving frame. Dirk Vdm's solution is more general. Note that his first equation is simply the derivative of Lorentz transform equation for t'. http://en.wikipedia.org/wiki/Lorentz_transformation -- --Bryan === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Shubee, did you try it derive it yourself? Bryan, I have derived the Lorentz transformation from the weakest axiom set possible. Trust me, I know that the Lorentz transformation says almost nothing by itself. See http://www.everythingimportant.org/relativity/special.pdf Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? >> This wording (time dilation *for* a clock-carrying traveler?) >> reveals a definitely sad amount of ignorance. I noticed that you repeated the same definitely sad amount of > ignorance below. Shubee > http://www.everythingimportant.org/relativity/special.pdf >> Relevant equation: >> Dt' = gamma ( Dt - v Dx ) >> Assumption: >> A clock is at rest in the unprimed frame, i.o.w. >> Dx = 0 >> Result: >> Dt' = gamma Dt >> whence >> Dt' > Dt >> meaning that in the unprimed frame the time between two ticks of >> the (moving) clock is dilated with respect to the time between those >> ticks as measured in the rest frame of the clock. >> General conclusion: >> Physics is not merely an application of algebra. Correct, which means that we should ignore your ignorant application > of algebra and complete misunderstanding of what the Lorentz > transformation implies. Very nice: http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/CompleteMisunderstand i ng.html Dirk Vdm === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=mV9EXQoAAACmCMM9qg0N4eJlXyr2Z93U 5.0),gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf None. === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? Shubeehttp://www.everythingimportant.org/relativity/special.pdf None. So how would you prove that the Lorentz transformation implies time dilation without presupposing more than just the Lorentz transformation? Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_and_Faraday.27s_law _of_induction Sue... Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=eFTX7goAAACJBRoHHOnDC9IuTZeZT1_H 1.1.4322),gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz force#Lorentz force and Faraday.... Sue... Shubee- Hide quoted text - - Show quoted text - I don't think you'll find the answer there - it concerns the Lorentz force - not the Lorentz transformation. The answer to the OP is that the Lorentz transformation is enough if both systems are inertial (no accelerations or strong gravity fields). === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_and_Faraday.... Sue... Shubee- Hide quoted text - - Show quoted text - I don't think you'll find the answer there - it concerns the Lorentz > force - not the Lorentz transformation. The answer to the OP is that > the Lorentz transformation is enough if both systems are inertial (no > accelerations or strong gravity fields). If there is no force in either system then there is nothing to affect a clock. <<...it is impossible to perform a physical experiment which differentiates in any fundamental sense between different inertial frames. By definition, Newton's laws of motion take the same form in all inertial frames. Einstein generalized this result in his special theory of relativity by asserting that all laws of physics take the same form in all inertial frames. >> http://farside.ph.utexas.edu/teaching/em/lectures/node108.html Sue... === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > If there is no force in either system then there is nothing > to affect a clock. <<...it is impossible to perform a physical > experiment which differentiates in any fundamental sense > between different inertial frames. By definition, Newton's > laws of motion take the same form in all inertial frames. > Einstein generalized this result in his special theory of > relativity by asserting that all laws of physics take > the same form in all inertial frames. >>http://farside.ph.utexas.edu/teaching/em/lectures/node108.html You are assuming that the Lorentz transformation implies the principle of relativity, which is a false assumption. Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) If there is no force in either system then there is nothing > to affect a clock. <<...it is impossible to perform a physical > experiment which differentiates in any fundamental sense > between different inertial frames. By definition, Newton's > laws of motion take the same form in all inertial frames. > Einstein generalized this result in his special theory of > relativity by asserting that all laws of physics take > the same form in all inertial frames. >>http://farside.ph.utexas.edu/teaching/em/lectures/node108.html You are assuming that the Lorentz transformation implies the principle > of relativity, which is a false assumption. The word Lorentz is not even on the page. <> http://www.aip.org/pt/vol-58/iss-11/p31.html http://scitation.aip.org/journals/doc/PHTOAD-ft/vol_58/iss_11/31_1.shtml Sue... Shubee === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=8AmUjAkAAADZJSnu31qnEczJ_Xx7CWjh CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0),gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz force#Lorentz force and Faraday.... Sue... Do we have to search this for the answer, Suzy? SNO-O-O-O-RE ..... === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_and_Faraday.... Sue... Do we have to search this for the answer, Suzy? SNO-O-O-O-RE ..... It is no more than knowing the rules of chess before offering comment on some particular strategy. http://farside.ph.utexas.edu/teaching/em/lectures/lectures.html http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/index.htm Sue... === Subject: Re: Can time dilation be computed with just the Lorentz transformation and no other assumptions? posting-account=8AmUjAkAAADZJSnu31qnEczJ_Xx7CWjh CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0),gzip(gfe),gzip(gfe) What assumptions, if any, must be added to the Lorentz transformation > in order to compute time dilation for a clock-carrying traveler? http://en.wikipedia.org/wiki/Lorentz force#Lorentz force and Faraday.... Sue... Do we have to search this for the answer, Suzy? æSNO-O-O-O-RE ..... It is no more than knowing the rules of chess before > offering comment on some particular strategy. Well, that lets me out. I'm just a draughts champion! :-) http://farside.ph.utexas.edu/teaching/em/lectures/lectures.htmlhttp://web.mi t .edu/8.02t/www/802TEAL3D/visualizations/light/index.htm Sue... === Subject: Inverse of Harmonic Numbers - Approximation Formula posting-account=ig899wkAAAAe9dQz7LKowWKRo9OO-X2B 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Last year David W. Cantrell presented a new asymptotic series for the inverse of H(x), x > 0. I found it yesterday, as I was interested in the subject: However, I liked the approximation formula he provided in an older thread, which was more suitable for programming in classic HP calculators: 1/2*( 1/Sqrt(-3*W(-1/(12*e^(2*(x-g))))) - 1 ) In an attempt to get a better accuracy for n as low as 1 or 2 I have introduced an inner term to the approximation: n ~ 1/2*( 1/Sqrt(-3*W(-1/(12*e^(2*(x - g + 1/(150/11*e^(4*x))))))) - 1 ) Evaluating this even more improved approximate inverse for x = H(1), H(2), and H(3), we obtain 1.00014451815, 2.00000046888, and 3.00000000101, respectively. Table for some values of x, as computed with help of my HP 33s calculator program: x n (actual n) ------------------------------------------------ 1.00 1.00022409778 (1) 1.50 2.00002207370 (2) 2.45 6.00000020420 (6) 2.93 10.0108471094 (10.0108470927) 3.50 18.0907442948 (18.0907442943) 4.00 30.1532900556 (30.1532900558) Of course, numerical solutions can be found by solving 'H=Psi(n+1) + . 577215664902' for n. The question is: are there more accurate formulas for this? Gerson W. Barbosa === Subject: Re: Notation for partial function Suppose X subseteq Y and f:X -> Z may I write f:Y --> Z with a special kind of arrow, or something, to indicate that f is > partial on Y? I am able to answer my own question. I thought I had seen such notation f:Y >--> Z if dom f subset Y, i.e. LaTeX's rightarrowtail. He calls Y the source of f. I also like his notation for the set maps induced by f. For the forward f_{|-} i.e. something like LaTeX's vdash as a subscript; and for the backward f^{-|} i.e. something like LaTeX's dashv as a superscript. -- He is not here; but far away The noise of life begins again And ghastly thro' the drizzling rain On the bald street breaks the blank day. === Subject: Re: Notation for partial function >> Suppose X subseteq Y and f:X -> Z may I write >> f:Y --> Z >> with a special kind of arrow, or something, to indicate that f is >> partial on Y? I am able to answer my own question. I thought I had seen such notation f:Y >--> Z if dom f subset Y, i.e. LaTeX's rightarrowtail. He calls Y the source > of f. That notation may be confusing, since many category theorists will read that as: f is an injective function from Y to Z. You should at least be sure to clearly identify what you mean when you use this notation. -- Jesse F. Hughes How come there's still apes running around loose and there are humans? Why did some of them decide to evolve and some did not? Did they choose to stay as a monkey or what? -Kans. Board of Ed member === Subject: Help solving A/ PI = b^2 + bd to find b where A and d are known Importance: Normal Hi I'm trying to calculate the length of the two axis of an ellipse from its area for a program I am writing. I know the area is given as A = PI * a * b, where a and b are the major and minor axis respectively. In this scenario however, A is known and a is defined as (b + d) where d is also some known variable (half the distance between the two focal points). Hence I can rewrite the function as: A / PI = b^2 + bd The value I am trying to calculate then (from the known A and d) is b. Unfortunately its been a long time since I've had to do any serious equation solving so frankly I haven't a clue how to solve this. Is it even possible from the information already known? Ben === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known > Hi > I'm trying to calculate the length of the two axis of an ellipse from > its area for a program I am writing. I know the area is given as A = > PI * a * b, where a and b are the major and minor axis respectively. > In this scenario however, A is known and a is defined as (b + d) > where d is also some known variable (half the distance between the > two focal points). Hence I can rewrite the function as: A / PI = b^2 + bd The value I am trying to calculate then (from the known A and d) is b. > Unfortunately its been a long time since I've had to do any serious > equation solving so frankly I haven't a clue how to solve this. Is it > even possible from the information already known? > Ben You have a quadratic equation: b^2 + b*d - A/pi = 0 Plug the coefficients into the quadratic formula to find the roots of the equation: http://en.wikipedia.org/wiki/Quadratic equation Choose the positive root. === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known Importance: Normal Greg, out... now that I see the quadratic equation formula it seems so obvious. Worst part is now I see it I even remember that from my college days. I guess it really is true the saying use it or loose it Ben >> Hi >> I'm trying to calculate the length of the two axis of an ellipse from >> its area for a program I am writing. I know the area is given as A = >> PI * a * b, where a and b are the major and minor axis respectively. >> In this scenario however, A is known and a is defined as (b + d) >> where d is also some known variable (half the distance between the >> two focal points). Hence I can rewrite the function as: >> A / PI = b^2 + bd >> The value I am trying to calculate then (from the known A and d) is b. >> Unfortunately its been a long time since I've had to do any serious >> equation solving so frankly I haven't a clue how to solve this. Is it >> even possible from the information already known? >> Ben You have a quadratic equation: b^2 + b*d - A/pi = 0 Plug the coefficients into the quadratic formula to find > the roots of the equation: http://en.wikipedia.org/wiki/Quadratic_equation Choose the positive root. === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known > Greg, out... now that I see the quadratic equation formula it seems so obvious. > Worst part is now I see it I even remember that from my college days. I guess it really is true the saying use it or loose it > Ben I wonder. Were you one of those who said I'll never need this during math classes? === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known Importance: Normal > Greg, >> out... now that I see the quadratic equation formula it seems so obvious. >> Worst part is now I see it I even remember that from my college days. >> I guess it really is true the saying use it or loose it >> Ben I wonder. Were you one of those who said I'll never need this during > math classes? Lol. Oh no I enjoyed math - just been about 8yrs since I've studied anything like it. === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known > Hi > I'm trying to calculate the length of the two axis of an ellipse from > its area for a program I am writing. I know the area is given as A = PI > * a * b, where a and b are the major and minor axis respectively. In > this scenario however, A is known and a is defined as (b + d) where d is > also some known variable (half the distance between the two focal > points). Hence I can rewrite the function as: A / PI = b^2 + bd Rewrite as b^2 + b * d - A / PI = 0. Solve with your favorite formula, the quadratic formula. Since we know b > 0, we want the positive solution. Therefore, we choose to add instead of subtract (rationale below). The answer will be: b = [-d + sqrt (d^2 + 4A / pi) ] / 2 Always guaranteed to be nonnegative since: d = sqrt(d^2) <= sqrt (d^2 + 4A / pi) and subtract d from all sides to get 0 <= sqrt (d^2 + 4A / pi). === Subject: Re: Help solving A/ PI = b^2 + bd to find b where A and d are known reply-type=response Importance: Normal Josh, You star, Ben >> Hi >> I'm trying to calculate the length of the two axis of an ellipse from its >> area for a program I am writing. I know the area is given as A = PI * a * >> b, where a and b are the major and minor axis respectively. In this >> scenario however, A is known and a is defined as (b + d) where d is also >> some known variable (half the distance between the two focal points). >> Hence I can rewrite the function as: >> A / PI = b^2 + bd Rewrite as > b^2 + b * d - A / PI = 0. Solve with your favorite formula, the quadratic formula. Since we know b > 0, we want the positive solution. Therefore, we choose to add instead > of subtract (rationale below). The answer will be: b = [-d + sqrt (d^2 + 4A / pi) ] / 2 Always guaranteed to be nonnegative since: d = sqrt(d^2) <= sqrt (d^2 + 4A / pi) and subtract d from all sides to get 0 <= sqrt (d^2 + 4A / pi). === Subject: Negative function (trigamma involved)? posting-account=BE2uVAoAAADrJGjKcy1HRoFcqqRy13TX Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) I am trying to show that the following function is always negative, for any m and n >0: f(m,n) := psi[1](n)*psi[1](n+m) + psi[1](m)*psi[1](n+m) - psi[1] (m)*psi[1](n) where psi[1](.) is the trigamma function (http://mathworld.wolfram.com/ TrigammaFunction.html). I can see numerically that it is so but can't figure out how to demonstrate it. Any hint would be very welcome! 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Olsson PL 2540 Kuttainen SE: 980 16 Karesuando === Subject: Re: tommy1729 set axioms <2542528.1216161622113.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On 14 Jul., 21:40, amy666 Ax Ay (x=y <-> Az (zex <-> zey)) 2) axiom of the empty set Presumably meaning > Ex Ay ~(yex) 3) axiom of pairing Presumably meaning > Ax Ay Ez At (tez <-> t=x v t=y) 4) axiom of union Presumably meaning > Ax Ey Az (zey <-> Et (zet & tex)) 5) non-standard elements within the set of real > numbers always have properties that correspond to > the > properties of infinitesimal and unlimited > elements. I didn't know that the set of real numbers > contains > non-standard > elements. > Do you have any idea of the meaning of any of the > terms used > especially in 5)? yes. Then please enlighten me: > What is the set of real number in your theory? > Just some complete ordered field? > Do you have an axiom that states the existence of > such a set? > How do you define nonstandard element of R? > Etc. etc. read the other replies , i answer this one to lwalke i believe. 6) this set theory is game theory in disguise , > whatever holds in game theory can be restated in > this > set theory. Is this an *attempt* to restate Conway numbers? no. this set theory is complete and consistant > apart > from solving equations. this set theory is the analogue of a computer > with > oo resourses (as should be since math is about > computation) 7) axiom of logic : x = [x] ( also if x = > empty ) > tommy1729 ps : it has been requested that i explicitly > gave > my axioms , now i did. |R|^2 = |R| is consistant with it. ... in the sense that ~(|R|^2 = |R|) cannot be > proved > from the above > axioms? > I just note that > |R|^2 = |R| > is also consistant with the axiom > 0) tommy1729 is stupid are we getting personal again hmm ?? i already proved |R|^2 = R many times by now !! But you cannot really do that from the axioms you > gave without > a proper defintion of the terms occurring here. > I'm getting problems if I simply use my definitions > of terms, > e.g. > |A|=|B| :<=> there exists a bijective map f:A->B > <=> there exists a set F such that all elements of F > are > (Kuratowsky) ordered pairs (a,b) with a in A and b in > B and > whenever (a,b) in F and (a,b') in F then b=b' and > whenever (a,b) in F and (a',b) in F then b=b' and > for all a in A exists b in B such that (a,b) in F and > for all b in B exists a in A such that (a,b) in F. I doubt that it can be shown from your axioms that > |R| = |R| holds (let alone |R|^2 = |R|). evidently Q = Q for all Q !!! are you saying Q can =/= Q ??? It depends. It is not necessarily the case that the = sign is the equal sign, but some prefer to view this as a matter of writing ||=|| for the clumsier There exists a bijection between and . (Compare with the = in cos(x) = 1 + O(x^2), which is a similar (and even wronger) abuse of notation). Since your theoretical framework hardly allows one to talk about the smallest ordinal a set can be bijected to, I don't see how you want to define a free-standing cardinality |R|; in fact I don't even see how you can define There exists a bijection... on your grounds. do i need to add as axiom Q = Q for all Q ?? No, you need to define |.| hagman === Subject: Re: tommy1729 set axioms <29219662.1216151863473.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Jul 14, 1:46 pm, amy666 ullrich , jesse f hughes and hagman. impressive hmm. No, it's sad that you think you've given clear > axioms, let alone that > you think you've defended anything. i dont have the illusion that they are clear at first sight. Except for the ones that restate known axioms of set theory, they're > far from clear in any sight. in fact the axioms only become a bit clear all together , individually they are vague. Please read up just a little to find out how modern axiomatic systems > work. but they are somewhat lets say descriptive , you really need to think about them. see them as a whole. You need to learn how modern axiomatic systems work. I don't need to > think about your slop; I already have a reading list of a lifetime of > mathematics to study, including alternatives to standard mathematics. > Such mathematics is informed, intellectually organized, and well > written. That's what you need to learn. Why are you so averse to informing yourself as to > some basic > mathematical logic by which you'd come to actually > understand > something about the axiomatic method? because things are not so well defined as commenly believed. However well defined things are or are not, you only deprive yourself > by not studying the basics of the subject. e.g. what is a set ? (1) If we had to, we can do all of set theory without ever mentioning > the word 'set'. The primitive of set theory is 'e' (read element > of). That (and '=', read is equal to) are not defined. All other > predicate and function symbols are defined. (2) Anyway, in a theory > such as Z set theory, a technical definition of 'is a set' can be > provided: First, we prove that there exists exactly one object that has no > members: E!xAy ~yex Then we define: 0=x <-> Ay ~yex Then we define: x is an element <-> Ey xey C is a class <-> (Ey yec v c=0) S is a set <-> (S is a class & Ey Sey) C is a proper class <-> (C is a class & ~C is a set) u is an urelement <-> ~u is a class > Erm, doesn't this deny the existence of urelements? Then u is an urelement <-> ~ u is a class <-> ~(Ey yeu v u=0) <-> ~(Ey yeu v Ay ~yeu) <-> ~(Ey yeu v ~Ey yeu) <-> ~ true hagman === Subject: Re: tommy1729 set axioms <29219662.1216151863473.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > First, we prove that there exists exactly one object that has > no > members: E!xAy ~yex Then we define: 0=x <-> Ay ~yex Then we define: x is an element <-> Ey xey C is a class <-> (Ey yec v c=0) S is a set <-> (S is a class & Ey Sey) C is a proper class <-> (C is a class & ~C is a set) u is an urelement <-> ~u is a class Erm, doesn't this deny the existence of urelements? Yes, in certain theories, it is a theorem that there are no urelements. In other theories, it is undetermined whether there are urelements, and in other theories there are urelements. > u is an urelement > <-> ~ u is a class > <-> ~(Ey yeu æv æu=0) > <-> ~(Ey yeu æv æAy ~yeu) > <-> ~(Ey yeu æv æ~Ey yeu) > <-> ~ true Right. To make it is undetermined whether there are urelements, we could have a primitive predicate 'is a set', then modify the axiom of extensionality: (x is a set & y is a set) -> (x = y <-> Az(zex <-> zey)) (Other axioms may require modification also.) Then to ensure that there are urelements, we could adopt an axiom: ExAy(~yex & ~x is a class) Or (with that aforementioned modified axiom of extensionality) to ensure that there are no urelements, we could adopt as an axiom (the axiom of purity) the negation of the above formula: ~ExAy(yex & ~x is a class) which is equivalent to Ax x is a class MoeBlee === Subject: Re: tommy1729 set axioms update <15184714.1216319617669.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Jul 16, 7:46 am, amy666 1) axiom of extensionality 2) axiom of the empty set 3) axiom of pairing 4) axiom of union 5) non-standard elements within the set of real > numbers always have properties that correspond to (snip) But even stronger, your axiom 8 entails that every > set has exactly one > member, so there is no 'set of real numbers' or > anything like that. NO !! you got it all wrong : its not true that a set can have only element. and not only Jack Markan but most here got it wrong and deduce other wrong things based on that misconception. lets repeat axiom 8 : x = [x] x may also be empty or denote a set itself. thus we have = [] = [[]] > and > x = [x] = [[x]] now lets consider [x] where this set does contain two elements. x = a,b then we get [x] = x = a,b = [a,b] and [a,b] clearly has 2 elements : a and b. you see ? perfectly consistant !! 6) axiom of regularity Without any form of comprehension axiom to prove the > existence of > intersections, I suppose 6 would be stated: ~x=0 -> EmexAy~(yem & yex) 7) axiom of infinity 8) axiom of logic : x = [x] ( also if x = empty ) I don't know why you call that 'axiom of logic'. Your system is inconsistent using just axioms 2, 3, > 8, and basic > identity theory: member of {S}. > So S is a member of S, which contradicts that S has > no members. And axiom 8 is inconsistent with and axioms 3 and 6: Let x={x}. So Az(Ay~yez -> ~x=z) (i.e., even without > extensionaliy, we > know that x is not an empty set since it has a > member, viz. x). So for > all m in x, there is no set that is in both m and in > x. But the only > member of x is x, so m=x. So x in m and in x, > contradicting that there > is no set that is in m and in x. Also, axiom 8 entails that that every set has exactly > one member. here you repeat yourself again ! you still dont understand that axiom 8 still allows sets with more than one member. and thus everything based on that misconception is a misconception. since you repeat your arguments in a single post , so will i : you got it all wrong : its not true that a set can have only element. and not only Jack Markan but most here got it wrong and deduce other wrong things based on that misconception. lets repeat axiom 8 : x = [x] x may also be empty or denote a set itself. thus we have = [] = [[]] > and > x = [x] = [[x]] now lets consider [x] where this set does contain two elements. x = a,b then we get [x] = x = a,b = [a,b] and [a,b] clearly has 2 elements : a and b. > Your argument is that [a,b] clearly has 2 elements. You never bothered to define most of your notation, but apparently the number of elements of something enclosed in square barackets is essentially simply the number of commas plus 1. Thus [a,b] has two elements. (What fun we can have in the special case a=b, I do not dare to imagine) And [x] has one element. Then your statement [x] = x = a,b = [a,b] claims that the same object has one element and two elements in the same time. hagman === Subject: Re: tommy1729 set axioms update > I think I might have used the name at Math Forum. Are you posting > through that interface? It might be that Math Forum is attaching my > old name to posts made from my current account made away from Math > Forum. Jack Markan is given as your name in the profile at Math Forum. It doesn't appear anywhere in the headers of your posts posted through Google. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: tommy1729 set axioms update <29391588.1216411570205.JavaMail.jakarta@nitrogen.mathforum.org> <87iqv07lkr.fsf@alatheia.dsl.inet.fi> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Jul 20, 6:05æam, Aatu Koskensilta posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > jack markan is your own name , genius !!! :p > I think I might have used the name at Math Forum. Are you posting > through that interface? It might be that Math Forum is attaching my > old name to posts made from my current account made away from Math > Forum. > and it is listed with every post you make. > It should not be. As I mentioned, are you posting through Math Forum? > Or do you only sometimes use Math Forum and other times some other > news interface? Because I notice now that sometimes you respond to my > posts as they are identified as by 'MoeBlee' and sometimes as they are > identified as 'Jack Markan'. You're right. Jack Markan appears at Math Forum. Indeed, several posters have different names at Math Forum from through other newsreaders, including Bob Kolker and Lester Zick. And this of course reignites the long standing debate between free newsreaders (Google Groups, Math Forum, Outlook Express, etc.) and the full-priced news servers. In particular, this will give proponents of the full-priced news servers another reaon to reject free newsreaders (and possibly killfile those who use them) -- namely, that they give incorrect user names. === Subject: Re: tommy1729 set axioms update <29391588.1216411570205.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > You're right. Jack Markan appears at Math Forum. MoeBlee === Subject: Re: tommy1729 set axioms update > And this of course reignites the long standing debate between > free newsreaders (Google Groups, Math Forum, Outlook Express, > etc.) and the full-priced news servers. Google Groups and Math Forum are not newsreaders. Outlook Express is, and is used with a news server of the users choice. Price is irrelevant; there are many free news servers. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: tommy1729 set axioms update > And this of course reignites the long standing debate between > free newsreaders (Google Groups, Math Forum, Outlook Express, > etc.) and the full-priced news servers. In particular, this will give > proponents of the full-priced news servers another reaon to reject > free newsreaders (and possibly killfile those who use them) -- > namely, that they give incorrect user names. It hasn't a damn thing to do with free news readers. It has to do with one strange news server. -- Jesse F. Hughes The future is a fascinating thing, and so is history. And you people are a fascinating part of history, for those in the future. -- James S. Harris is fascinating, too === Subject: Re: tommy1729 set axioms update <10493260.1216219611966.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > On Jul 17, 6:56æam, Aatu Koskensilta -- e.g. in the form of Dedekind cuts -- are sets, and we can, with > some care, reformulate most of analysis in the theory, the same way we > do in e.g. ACA 0. In effect, the collection of the reals is treated as > a proper class. > lwal). Moreover, I noticed that that theory has a quite powerful > impredicative-over-proper-classes comprehension schema; so it's hardly > akin to Tommy's fantasy in which he has no comprehension (not even > separation or replacement, etc.) axioms at all, as well as no power > set axiom. You got me there. PST has a comprehension schema, whereas TST -- at least the version of TST in this thread -- has no schemata at all. === Subject: Point Torus Have a balloon and point with your fingers at opposite sides until they touch each other. What is the genus of this surface? With one vertex in that point you can draw a complete graph on it with 5 vertices (a K5), but not a K6. If you want to triangulate all faces of the K5 you have double edges. Using Heawoods formula n(g)=(7+sqrt(1+48g))/2 for the number n of colours we need it should have genus 1/6? 1/6 of a hole? Or is this a non legal surface? === Subject: dual ellipsoid posting-account=RcU6uwoAAABeyBZDbjLICsz3Bd43yvlG SV1),gzip(gfe),gzip(gfe) How is dual ellipsod defined, what is it's relation to the original === Subject: Re: dual ellipsoid- > How is dual ellipsod defined, what is it's relation to the original Imagine the unit sphere in R^3: x^2 + y^2 + z^2 = 1. Take an arbitrary point (x1, y1, z1) <> (0, 0, 0). Consider the plane x.x1 + y.y1 + z.z1 = 1. In classical analytic geometry one considers this both as the point set {(x,y,z)|x.x1 + y.y1 + z.z1 = 1} and as a mathematical entity in its own right, a thing named plane and characterised by the triple (x1, y1, z1). In this way a one-to-one mapping of points different from O = (0, 0, 0) onto planes not passing through O is established. This mapping and its inverse are together denoted as a polarity with respect to the quadric x^2 + y^2 + z^2 = 1. Such pairs of points and planes are named pole and polar. One can play this game with any non-degenerate quadratic polynomial in any number of real variables. The game is played far better in projective space: then the origin O and planes through O no longer play an exceptional role; O and the plane at infinity correspond to each other, as do planes through O and points at infinity. But that does not bother us for now. Now for the dual ellipsoid. A main theorem in classical analytic geometry states that whenever the pole P lies in a quadric ( = quadratic surface) A its polar plane is tangent to another quadric B. Furthermore, this relation is reciprocal: the pole of any plane tangent to A lies in B. Let the given ellipsoid have half-axes A, B, C. Take its principal axes as coordinate axes. Then its equation is the well-known (x/A)^2 + (y/B)^2 + (z/C)^2 = 1. It is not difficult to deduce that its dual (with respect to the unit sphere) has half-axes 1/A, 1/B, 1/C and has the equation (Ax)^2 + (By)^2 + (Cz)^2 = 1. In analytical dynamics vital roles are played by the ellipsoid of inertia and its dual. Consult for instance: 7TH EDITION 1905 or any other edition. Next to nothing to be found in Wikipedia; more at http://mathworld.wolfram.com/Polar.html Ciao: Johan E. Mebius === Subject: Convergent series for Exponential integral posting-account=3M4yTgoAAABvI2Q_4aJrDHIFHGf0Ul_I Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hi I need a little help here. I have a function, f = exp (i*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2). I need to integrate this function with respect to x. As this is an exponential integral, exact integral is not possible. what could be the convergent series for this? === Subject: Re: Convergent series for Exponential integral posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Hi > I need a little help here. > I have a function, æf = exp (i*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2). I need to integrate this function with respect to x. > As this is an exponential integral, exact integral is not possible. > what could be the convergent series for this? Asking sci.math would be better for this type of thing. My suggestion is expand f in a power series, then integrate term by term. I cannot promise convergence though. However if you want to integrate along (-infty,infty) or [0,infty) I believe this can be solved exactly using contour integration. === Subject: Re: Convergent series for Exponential integral > Hi > I need a little help here. > I have a function, f = exp (i*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2). I need to integrate this function with respect to x. > As this is an exponential integral, exact integral is not possible. > what could be the convergent series for this? Maple V release 4 gives (assuming a > 0): > f :=x-> exp (I*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2); > series(f(x),x); > assume(a>0); > simplify(); > int(,x); exp(I k a~) I (I exp(I k a~) + a~ k exp(I k a~)) 3 ----------- x + 1/6 ------------------------------------ x + 1/5 ( a~ 3 a~ 2 2 3/8 exp(I k a~) - 3/8 I exp(I k a~) k a~ - 1/8 exp(I k a~) k a~ / 5 5 7 ) / a~ x + O(x ) / -- I.N. Galidakis === Subject: Re: Convergent series for Exponential integral posting-account=3M4yTgoAAABvI2Q_4aJrDHIFHGf0Ul_I Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) i = sqrt(-1) and k,a are constants. === Subject: ANN: Auguri posting-account=JFOcoQoAAABmf-SaDlEwVMK52aS1sH9x CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) For Immediate Release Contact: Advanced Analytics Group, promo@aag-auguri.com Statistical, Frequency, Linear and Nonlinear Analysis, and Forecasting Powerhouse Advanced Analytics Group has released Auguri 2.1, an integrated Windows data exploration, analysis, and forecasting tool with emphasis on nonlinear dynamical methods. Auguri provides tools for the manipulation and analysis of data multivariate model design, and from the specification of solutions to these models to their comparison and storage for later use, under a powerful and intuitive graphical user interface. 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Define solutions according to your preferences and data characteristics. Compare auto- regressive with local least squares methods side-by-side, or against powerful multilayer feedforward neural networks trained with innovative quasi-unsupervised learning algorithms. Any and all is a choice. No restrictions apply. Prices begin at $549.00 (US), with deeply-discounted academic licenses. # # # 30 day Evaluation Copy Available from http://aag-auguri.com/download.html # # # We are also looking for individuals who are willing to submit a case study using our software in exchange of a free license and a possible monetary compensation (US $200.00 ) if we choose to publish your submission. If interested, and require an extended trial license, please reply to promo@aag-auguri.com. === Subject: Cancel Re: A consideration concerning the diagonal argument of G. Cantor(was Q) -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: I hate to ask, but I must - 3 coloring vertex problem > What's your thoughts on > www.geocities.com/dharwadker/vertex_coloring Hard to say. On a fairly modest 255 vertex graph, on a machine with 2GB of memory, it maxed out the swap space, took over the machine, and got killed by the system monitor. Makes it hard to evaluate for real.