mm-3459 === Subject: Re: An Invitation to Quantum Mathematics I'm just guessing here, but it seems like this is getting into an area of abstract algebra based on the ideas of Von Neuman et al, numbers and operators as objects, etc. But I dont think that it answers the question. Of course, one would need to know some abstract algebra, which I dont, and then one would wonder why the solution to the central riddle of the entire cosmos rates about 20 hits on Google. There is a very simple explanation of whichway information which can be are merely probabilistic deformations of that probabilistic manifold i.e. they are just probabilities in the first place, hence when chopped into trivially sized fragments they remain existent in the form of a probabilistic entity due to conservation, when uncertainty is destroyed so (ahem) energy. If you know of an abstract algebra explanation of whichway info please post a ref. And you know something, speaking of Harris Escher Bach, these silly trivialistic ideas really are trivial. A theory based on trivia is just that - trivial. However, existence of a trivial is indeterminate, and so validity of such a theory is not negated but merely indeterminate - which is exactly what one would expect physics to look like if Godel is to be believed. A theory of physics which is known to be inherently crippled, and it is exactly as it should be, because the universe is not determined. HAH ! The mind reels at the prospect, but how could anything else possibly be true. > Properties which hold for Pisier Spaces, can be Google up by searching > for Pisier Space. J.S.Harris Fan schreef: Does this allow you to model whichway information experiments ? > For all examples of Hilbert Spaces, their dual quantum spaces turned > out to be different from the initial spaces. Piser showed that amound > all quantum Hilbert spaces there is exactly one behaving in complete > correspondence with the classical Riesz theorem: for this space its > canonical bijection is a completely isometric conjugate linear > operator. Thus, such quantum space plays in quantum mathmatics the same > role as the usual Hilbert Space does in the classical functional > analysis. markwh04@yahoo.com schreef: Quantum Mathematics is the mathematical apparatus of quantum > mechanics. > What is the essence of this mathematical ideology ? It is... * Classical Mathematics deals [sic] exclusively with spaces of > functions and > its main structure is the uniform norm. > * Quantum Mathematics deals with the spaces of operators and the main > structure is the quantum norm. Not any of these. There is a Hilbert space formulation for Classical > Mechanics, as well as Quantum Mechanics; or any other physical theory > whose observables form a C*-algebra. There is a phase space representation for quantum mechanics, with > states represented by *positive* phase space distributions. These > distributions are Gaussian convolutions of the Wigner functions > representing the states and can alternatively be characterized as > transition probabilities with respect to the coherent states. Hence if > W is a mixed state with density matrix rho, then = > rho(p,q) is the phase space density corresponding to W, where |p,q> is > the coherent state for phase space point (p,q). This function rho(p,q) > is the Gaussian smear of the Wigner function W(p,q). Though W(p,q) is > not positive definite, rho(p,q) will always be. Another way of saying this is that quantum distributions in phase > space (that is: Wigner functions) are inverse Gaussian convolutions of > phase space densities. So classical mathematics applies to quantum mechanics, quantum > mathematics applies to classical mechanics. There is even a classical analogue of the celebrated Naimark theorem. > The only difference is that the Naimark extension of a classical > system is generally a quantum system (which, after the fact, justifies > using quantum mechanics techniques even for classical physics) === Subject: Re: An Invitation to Quantum Mathematics Just one bit of clarification is in order - > There is a very simple explanation of whichway information which can be > stated in one sentence as follows. Length itself is probabilistic, > are merely probabilistic deformations of that probabilistic manifold i.e. > they are just probabilities in the first place, hence when chopped into > trivially sized fragments they remain existent in the form of a > probabilistic entity due to conservation, when uncertainty is destroyed so > (ahem) energy. otherwise smooth probability gradient) Trivially small fragments of a you get trivial fragments. Theorem : The existence of a trivial is indeterminate. So, they simultaneously exist, and do not exist. It's indeterminate. the reach of physics, and hence their structure is indeterminate, in fact - undeterminable. the world of empiricism because they exist as uncertainties. Destroy the Godel, Escher, Bach ..... Rod Serling - === Subject: Re: An Invitation to Quantum Mathematics > Just one bit of clarification is in order - > There is a very simple explanation of whichway information which can be > stated in one sentence as follows. Length itself is probabilistic, > are merely probabilistic deformations of that probabilistic manifold i.e. > they are just probabilities in the first place, hence when chopped into > trivially sized fragments they remain existent in the form of a > probabilistic entity due to conservation, when uncertainty is destroyed > so > (ahem) energy. otherwise smooth probability gradient) Trivially small fragments of a unsplittable, > you get trivial fragments. > Harris' > Theorem : The existence of a trivial is indeterminate. So, they > simultaneously exist, and do not exist. It's indeterminate. > beyond > the reach of physics, and hence their structure is indeterminate, in fact - > undeterminable. > on > the world of empiricism because they exist as uncertainties. Destroy the Godel, Escher, Bach ..... Rod Serling they are more amenable to mathematical formalisms such as Harris' Theorem, which also leads directly to a discussion of order and disorder. No such consequences can be gleaned from virtuality - at least none that I'm aware of. I am unaware of a formal definition of virtualness. === Subject: Re: An Invitation to Quantum Mathematics Just one bit of clarification is in order - > There is a very simple explanation of whichway information which can be > stated in one sentence as follows. Length itself is probabilistic, > are merely probabilistic deformations of that probabilistic manifold > i.e. > they are just probabilities in the first place, hence when chopped into > trivially sized fragments they remain existent in the form of a > probabilistic entity due to conservation, when uncertainty is > destroyed > so of > (ahem) energy. otherwise smooth probability gradient) Trivially small fragments of a the > unsplittable, > you get trivial fragments. Harris' > Theorem : The existence of a trivial is indeterminate. So, they > simultaneously exist, and do not exist. It's indeterminate. beyond > the reach of physics, and hence their structure is indeterminate, in > fact - > undeterminable. on > the world of empiricism because they exist as uncertainties. Destroy the Godel, Escher, Bach ..... Rod Serling > they are more amenable to mathematical formalisms such as Harris' Theorem, > which also leads directly to a discussion of order and disorder. No such > consequences can be gleaned from virtuality - at least none that I'm aware > of. I am unaware of a formal definition of virtualness. Apparently, Wikipedia has an interesting definition of Virtual ----------from Wikipedia ----------from Wikipedia like someone's got reality confused with mathematics. Rocks are real. Abstract rocks are not. I though that this had been settled with Alexis Meinong. Personally, I think that notion of triviality is far more sensible than the notion of virtuality. Are we to believe that there is a bridge between the abstract and the real which lies outside of the confines of the mind ? If you bought that bridge I got one I can sell ya too. I'll sell ya one real cheap. Triviality, on the other hand, has a very sensible physical interpretation given the limitations imposed on space by the presence of Plancklength. Everybody knew about it for years. They should just admit it already.` === Subject: Re: An Invitation to Quantum Mathematics <1IqdnWf8IOkQ4sPYnZ2dnUVZ_uqdnZ2d@comcast.com> <1YGdnZjY3eXoDMPYnZ2dnUVZ_vqdnZ2d@comcast.com> There is a very simple explanation of whichway information which can > be > stated in one sentence as follows. Length itself is probabilistic, > are merely probabilistic deformations of that probabilistic manifold > i.e. > they are just probabilities in the first place, hence when chopped > into > trivially sized fragments they remain existent in the form of a > probabilistic entity due to conservation, when uncertainty is > destroyed > so > of > (ahem) energy. otherwise smooth probability gradient) Trivially small fragments of a > the > unsplittable, > you get trivial fragments. Harris' > Theorem : The existence of a trivial is indeterminate. So, they > simultaneously exist, and do not exist. It's indeterminate. beyond > the reach of physics, and hence their structure is indeterminate, in > fact - > undeterminable. on > the world of empiricism because they exist as uncertainties. Destroy the Godel, Escher, Bach ..... Rod Serling > they are more amenable to mathematical formalisms such as Harris' Theorem, > which also leads directly to a discussion of order and disorder. No such > consequences can be gleaned from virtuality - at least none that I'm > aware > of. I am unaware of a formal definition of virtualness. > like someone's got reality confused with mathematics. Rocks are real. > Abstract rocks are not. I though that this had been settled with Alexis > Meinong. Personally, I think that notion of triviality is far more sensible than the > notion of virtuality. Are we to believe that there is a bridge between the abstract and the real > which lies outside of the confines of the mind ? If you bought that bridge I > got one I can sell ya too. I'll sell ya one real cheap. Triviality, on the other hand, has a very sensible physical interpretation > given the limitations imposed on space by the presence of Plancklength. > Everybody knew about it for years. They should just admit it already.` === Subject: Re: An Invitation to Quantum Mathematics I think that the probabilistic route is much easier to understand. Modern algebra makes my head spin. But I did find something interesting about monodromy groups and transformations, etc. My main interest is this is understanding indeterminacy, randomness, causality, etc. Seems like it always has something to do with uniqueness being violated, and I'm guessing that this is probably also the case in modern algebra but much more difficult to make any sense of it, for me anyway. The monodromy theorem says something about the uniqueness of analytic continuation, so uniqueness is a central idea here, And also on monodromy (from Wikipedia) ------------------------- .....the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity..... ------------------------- so, indeterminacy is buried pretty deeply in the modern approach. Sort of like how it's deeply embedded in Axiom of Choice. I think that triviality is perhaps the most direct route to indeterminacy, discussion of triviality or it's equivalent it starts to sound like Alexis Meinong - unless of course I'm confusing nomenclature with English. me > like someone's got reality confused with mathematics. Rocks are real. If you search for Bogoliubov transformation you will see there is > experimental physical evidence for these virtual paticles. So there > proven. See eg. VOLUME 88, NUMBER 6 PHYSICAL REVIEW LETTERS 11 FEBRUARY > 2002: Experimental Observation of the Bogoliubov Transformation for a > Bose-Einstein Condensed Gas. > Just one bit of clarification is in order - > There is a very simple explanation of whichway information which can > be > stated in one sentence as follows. Length itself is probabilistic, > are merely probabilistic deformations of that probabilistic manifold > i.e. > they are just probabilities in the first place, hence when chopped > into > trivially sized fragments they remain existent in the form of a > probabilistic entity due to conservation, when uncertainty is > destroyed > so conservation > of > (ahem) energy. > of an > otherwise smooth probability gradient) Trivially small fragments of a to > the > unsplittable, > you get trivial fragments. Harris' > Theorem : The existence of a trivial is indeterminate. So, they > simultaneously exist, and do not exist. It's indeterminate. > are > beyond > the reach of physics, and hence their structure is indeterminate, in > fact - > undeterminable. > felt > on > the world of empiricism because they exist as uncertainties. Destroy the Godel, Escher, Bach ..... Rod Serling because > they are more amenable to mathematical formalisms such as Harris' Theorem, > which also leads directly to a discussion of order and disorder. No such > consequences can be gleaned from virtuality - at least none that I'm > aware > of. I am unaware of a formal definition of virtualness. > Apparently, Wikipedia has an interesting definition of Virtual ----------from Wikipedia some respect > ----------from Wikipedia me > like someone's got reality confused with mathematics. Rocks are real. > Abstract rocks are not. I though that this had been settled with Alexis > Meinong. Personally, I think that notion of triviality is far more sensible than the > notion of virtuality. Are we to believe that there is a bridge between the abstract and the real > which lies outside of the confines of the mind ? If you bought that bridge I > got one I can sell ya too. I'll sell ya one real cheap. Triviality, on the other hand, has a very sensible physical interpretation > given the limitations imposed on space by the presence of Plancklength. > Everybody knew about it for years. They should just admit it already.` > === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... How about f_1 = f_2 = f_3 = 0? -- -kira === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... d/dx f1 = C1 d/dx f2 >d/dx f2 = C2 d/dx f3 >d/dx f3 = C3 d/dx f1 >where C1,C2,C3 are constants, f1,f2,f3 are continuous? >It would be nice if f1,f2,f3 are bounded. > This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which > case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), > the only possible solution is f1 = constant, which implies f2 and f3 > are also constant. -- > Daniel Mayost Sorry, what I meant was, d/dx f1 = C1 f2 d/dx f2 = C2 f3 d/dx f3 = C3 f1 === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous? >>It would be nice if f1,f2,f3 are bounded. >> This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which >> case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), >> the only possible solution is f1 = constant, which implies f2 and f3 >> are also constant. >> -- >> Daniel Mayost Sorry, what I meant was, >d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 > In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. -- Daniel Mayost === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous? >>It would be nice if f1,f2,f3 are bounded. >> This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which >> case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), >> the only possible solution is f1 = constant, which implies f2 and f3 >> are also constant. >> -- >> Daniel Mayost Sorry, what I meant was, >d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 > In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear > ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), > then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. There is no (easy) way to bound f1,f2,f3? -- > Daniel Mayost === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous? >>It would be nice if f1,f2,f3 are bounded. >> This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which >> case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), >> the only possible solution is f1 = constant, which implies f2 and f3 >> are also constant. >> -- >> Daniel Mayost Sorry, what I meant was, >d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 > In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear > ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), > then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. There is no (easy) way to bound f1,f2,f3? Let's assume that C1, C2, and C3 are real. Given some initial conditions for these functions, you have real initial values (say at x = 0) for f1 & its first and second derivatives there, together with the ODE f1''' = k^3 f1. It is then possible to solve for the exact solution. If k = 0, then the solution is a polynomial of degree (at most) 2. If k < 0, then solutions decay exponentially, but if k > 0, we could have exponential growth. Short answer: Fairly easy sharp bounds if you know what C1, C2, and C3 are and if you have and initial value for the functions. -- chip === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous? >>It would be nice if f1,f2,f3 are bounded. >> This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which >> case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), >> the only possible solution is f1 = constant, which implies f2 and f3 >> are also constant. >> -- >> Daniel Mayost Sorry, what I meant was, >d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 > In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear > ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), > then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. There is no (easy) way to bound f1,f2,f3? > Let's assume that C1, C2, and C3 are real. Given some initial conditions for these functions, > you have real initial values (say at x = 0) for f1 & > its first and second derivatives there, together > with the ODE f1''' = k^3 f1. It is then possible > to solve for the exact solution. If k = 0, then the > solution is a polynomial of degree (at most) 2. If k < 0, then solutions decay exponentially, > but if k > 0, we could have exponential growth. but exp() is only bounded on one side... when you take integration, it still blows up right? Short answer: Fairly easy sharp bounds if > you know what C1, C2, and C3 are and if > you have and initial value for the functions. -- chip === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous? >>It would be nice if f1,f2,f3 are bounded. >> This reduces to f1' = C1 C2 C3 f1'. So unless C1 C2 C3 = 1 (in which >> case f1 can be anything and then f2=f1/C1+a and f3=f1/(C1 C2)+b ), >> the only possible solution is f1 = constant, which implies f2 and f3 >> are also constant. >> -- >> Daniel Mayost Sorry, what I meant was, >d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 > In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear > ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), > then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. There is no (easy) way to bound f1,f2,f3? > Let's assume that C1, C2, and C3 are real. Given some initial conditions for these functions, > you have real initial values (say at x = 0) for f1 & > its first and second derivatives there, together > with the ODE f1''' = k^3 f1. It is then possible > to solve for the exact solution. If k = 0, then the > solution is a polynomial of degree (at most) 2. If k < 0, then solutions decay exponentially, > but if k > 0, we could have exponential growth. but exp() is only bounded on one side... > when you take integration, it still blows up right? It blows up at infinity. Did you expect a solution that remains bounded on the real line? It's a homogenous system, so in the absence of initial conditions, you always have the trivial soluions, f = 0. This is the only solution that remains bounded over R. You have not given enough information about C1, C2, C3 (or about initial conditions) to make clear sense of the wish to have bounds. === Subject: Re: Are there functions f1(x) f2(x) f3(x), such that... d/dx f1 = C1 f2 >d/dx f2 = C2 f3 >d/dx f3 = C3 f1 In that case, this reduces to f1''' = C1 C2 C3 f1, which is a basic linear > ODE with constant coefficients. So if you let k = (C1 C2 C3)^(1/3), > then f1 can be any linear combination of: exp(kx) exp(-kx/2) cos(sqrt(3) kx/2) > exp(-kx/2) sin(sqrt(3) kx/2) with f2 and f3 following easily. There is no (easy) way to bound f1,f2,f3? > Yes, most easy. f1(x) = f2(x) = f3(x) = 0. === Subject: Need help with Abel's convergence theorem (Abel's Theorem) Given that SUM(0,infty)a_k is convergent, a_k in C (or equivalently that the convergence set of SUM(0,infty)a_kz^k contains Delta(0;1) U {1}), let r>=1, and let E={|1-z|<=r(1-|z|)}. We have E a subset of Delta(0;1) U {1}. Show that f is continuous at 1, f being the sum of SUM(0,infty) a_kz^k on E. Sketch E for r=1, and r>1. I am trying to work out the significance of the set E={|1-z|<=r(1-|z|)}? How does this fit into the theorem? === Subject: Re: Need help with Abel's convergence theorem >(Abel's Theorem) Given that SUM(0,infty)a_k is convergent, a_k in C (or >equivalently that the convergence set of SUM(0,infty)a_kz^k contains >Delta(0;1) U {1}), let r>=1, and let E={|1-z|<=r(1-|z|)}. We have E a >subset of Delta(0;1) U {1}. Show that f is continuous at 1, f being the >sum of SUM(0,infty) a_kz^k on E. Sketch E for r=1, and r>1. I am trying to work out the significance of the set >E={|1-z|<=r(1-|z|)}? How does this fit into the theorem? Have you figured out what E looks like? It's a nontangential approach region - a certain subset of the disk which near the point 1 looks like the region bounded by two rays starting at 1 and pointing into the disk at a certain angle. How it fits into the theorem is that it's possible to show that f _is_ continuous on E; this says that the sum has a limit as you approach 1 from the interior of the disk in a non- tangential way. If you allow arbitrary approach to 1, for example along a curve in the open disk which is _tangent_ to the unit circle at 1, then the theorem becomes false. Possibly you meant to ask how the definition of the region E fits into the _proof_ of the theorem. That would raise the question have you got any ideas for how the proof might go... ************************ === Subject: Re: Need help with Abel's convergence theorem (Abel's Theorem) Given that SUM(0,infty)a_k is convergent, a_k in C (or >equivalently that the convergence set of SUM(0,infty)a_kz^k contains >Delta(0;1) U {1}), let r>=1, and let E={|1-z|<=r(1-|z|)}. We have E a >subset of Delta(0;1) U {1}. Show that f is continuous at 1, f being the >sum of SUM(0,infty) a_kz^k on E. Sketch E for r=1, and r>1. I am trying to work out the significance of the set >E={|1-z|<=r(1-|z|)}? How does this fit into the theorem? Have you figured out what E looks like? It's a nontangential > approach region - a certain subset of the disk which near the > point 1 looks like the region bounded by two rays starting > at 1 and pointing into the disk at a certain angle. How it fits into the theorem is that it's possible to show that > f _is_ continuous on E; this says that the sum has a limit > as you approach 1 from the interior of the disk in a non- > tangential way. If you allow arbitrary approach to 1, > for example along a curve in the open disk which is > _tangent_ to the unit circle at 1, then the theorem > becomes false. Possibly you meant to ask how the definition of the > region E fits into the _proof_ of the theorem. That > would raise the question have you got any ideas for > how the proof might go... > ************************ what the theorem was actually saying before). In answer to your question, the author is kind enough to offer a hint: f(1) - f(z) = SUM(0,infty)a_k(1-z^k) = SUM(1,m)a_k(1-z^k) + SUM(m+1,infty)a_k(1-z^k) Then apply the Abel identity to SUM(m+1,m+p)a_k(1-z^k). I'll give it a go! -Darren === Subject: Isotherms in a unit circle with equation x^2+y^2<1 or =1 The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 - y +10. (a) Sketch the isotherm T(x,y)= 10 (b) Find the hottest and coldest points on the plate --------------------------------------------------------------------------- How could the problem above be solved. How to interpret the results of partial differentiation of the above function. Could I use gredient to look for the maximum increase, but that would not tell me the temperature is hot or cold. Please help me! === Subject: Re: Isotherms in a unit circle with equation x^2+y^2<1 or =1 > The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 > - y +10. > (a) Sketch the isotherm T(x,y)= 10 > (b) Find the hottest and coldest points on the plate plot3d can be used to directly plot it by graphics.Contour plots give loops of same temperature. === Subject: Re: Isotherms in a unit circle with equation x^2+y^2<1 or =1 > The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 > - y +10. > (a) Sketch the isotherm T(x,y)= 10 > (b) Find the hottest and coldest points on the plate Numerically plot3d can be directly used to plot by graphics.Contour plots gives loops of same temperature. === Subject: Re: Isotherms in a unit circle with equation x^2+y^2<1 or =1 > The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 > - y +10. > (a) Sketch the isotherm T(x,y)= 10 > (b) Find the hottest and coldest points on the plate Numerical plot3d can be directly plotted by graphics.Contour plots gives loops of same temparture. === Subject: Re: Isotherms in a unit circle with equation x^2+y^2<1 or =1 > The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 > - y +10. > (a) Sketch the isotherm T(x,y)= 10 > (b) Find the hottest and coldest points on the plate Numerical plot3d can be directly plotted by graphics.Contour plots gives loops of same temparture. === Subject: Re: Isotherms in a unit circle with equation x^2+y^2<1 or =1 bottomline ha escrito: > The Temperatures at any point P(x,y) on the plate is T(x,y)= 2x^2 + y^2 > - y +10. > (a) Sketch the isotherm T(x,y)= 10 Resolve T(x,y)= 10 T(x,y)= 2x^2 + y^2 - y +10= 10, 2x^2 = y - y^2 and finaly sketch x = ROOT((y - y^2)/2), of course consider both roots > (b) Find the hottest and coldest points on the plate Check the second derivate, if is < 0 is maximum, > 0 is minimum > --------------------------------------------------------------------------- > How could the problem above be solved. How to interpret the results of > partial differentiation of the above function. Could I use gredient to > look for the maximum increase, but that would not tell me the > temperature is hot or cold. Also you can get graphicaly by drawing hotter or cooler isotherm, try T(x,y)=11,12,... until you get the idea where the hottestcoolest point. I hope it helps > Please help me! alen Isenhart u#440483 my game: http://uc1.gamestotal.com/?tft=de53 === Subject: Why Regularity? The lier paradox arose from our definition of M as either a lier or a truth teller, in the following manner. M is a lier<->M:As by M -> s is ~true. M is a truth teller<->M:As by M -> s is true. A for every and s means 'statement. Now, if s= I always say the truth , then what we will have is a tautology. were M exists but we don't know weather M is a lier or a truth teller( the truth teller non paradox ), this means that if we have further information about M, then this further information will decide weather M is a truth teller or a lier! Now,if s= I never say the truth , here we have a contradiction about M, is M a lier or a truth teller, since for such s, M is a lier <-> M is a truth teller, A contradiction ( The lier paradox), therefore M doesn't exist. Now by analogy with sets M={s|s!es} raise the lier paradox, while M={s|ses} raise the truth teller non paradox. Then Why regularity? It appears to me that when we state that s is not in s , then we are making a LIE. But A theory which has the axiom of regularity appears consistent because such theory would be like The man who always lies. This man is a man that we can work with in a consistent manner, we simply can know the truth value from his statements by simply reversing them. That's why ZFC has some consistency. But yet it couldn't have a universe, because every set in it is a LIE, and the set of all lies do not exist, since it is contradictive. It appears to me , in order to derive a more consistent set theory, we should adopt exactly the opposite way, i.e. an axiom of Irregularity. Let us try build the following set theory:- 1) Axiom of Extentiality. 2) Axiom of Irregularity: Ax xex 3) Axiom of Uniqueness: EU,Ay,Az (zey <-> z=y) -> y=U There exist a unique set U that for every y, every z, zey <-> z=y , then y=U. 4) Axiom of Pairing. 5) Axiom of Union. 6) Axiom of Separation. 7) Axiom of Replacement. 8) Axiom of Power set. 9) Axiom of Choice. All axioms except 2) and 3) are as in ZFC. Now what is against this theory. This theory has a universe, and it is in itself. And avoids Russell's paradox all together, and I assume that even the paradoxes of the ordinal of all ordinals, and the cardinality of all cardinals, are avoided in this theory. There is even no need to change the ordinary set notation of this theory. For example the pairing of x and y according to this theory is {x,y,{x,y}} , but for short there is no need to mention the member {x,y}, we can write it as {x,y} since it is understood from Ax.2 that {x,y} is in that set, lets say that the later member is invisible. Likewise the intersection of two sets A={ x,y ,{x,y} } and B={x,z,{x,z}} is in reality A.B={x,A.B} however there is no need to mention A.B as a member, so we can write it A.B={x} However in this theory there is no empty set, but there is an irregular singlton set that is unique. so A={A} were A is singlton -> A is a unique set in this theory, and according to 3) A=U. Therefore for example K={x,K} and L={y,L} then K.L = {K.L} And for J={s,J} and Q={p,Q} then J.Q={J.Q} According to this theory J.Q = K.L = U. Accordingly U is a subset of every set in this theory. Now , can anybody tell me what is wrong with this theory. Is it inconsistent? Zuhair === Subject: Re: Why Regularity? The reason why we have the axiom of regularity isn't really anything to do with the liar paradox. It's just an indication of the fact that we only want to study well-founded sets. In ZF-regularity we can prove that the class of well-founded sets is a model for ZF with regularity anyway. === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of finding consistency of statements of the that who always lies. All sets in ZFC are lies. Yet , I said such M that all of its statements are lies do have consistency, it is the consistency of the total lier, though I call the later ( the opposite truth teller ) anyhow. This is philosophical. Putting the axioms of regularity confines us to the lies that are fairly consistent, it makes us avoid the lier paradox, which reveal the philosophical truth to ZF. ZFC with Regularity is consistent logically, but philosophically speaking, it is a system of consistent lies, see the ulternative that I have proposed, I think this is the HONEST set theory . But It seems not so practical. The Lieying ZF ( the standard one ) is easier to deal with. And since all what we require from a set theory is consistency, then it doesn't really matter if ZF is philosophically a consistent lie. Since it easier to deal with, then let it. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved that no language can define its own truth predicate. That's how the liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of regularity. === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. Tarski is wrong! see set theory I have proposed, it has no problem with a universe. The set of all things in thereselfs, defines its own truth predicate. see the set theory I have proposed, were I need you to formulate the infinity axiom in it, that is if you are willing to help me. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Read about regularity only, it is about the whole of ZFC, the theorym of consistent lies. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. > Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless gibberish. === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read something you have to think about it, not just refuse it straightforwards. There is great similarity between Russells sets and the Lier paradox, you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted comprehension on ZF, and remove Regularity, then you will see ZF going to its true nature. Just putting a restriction like regularity, though avoids this paradox, but doens't let you reveal the truth of this theory. IF you are driving your car in the wrong direction, and you discovered that, and reduced your spead so that you will not reach the wrong aim, it doesn't mean that you are not driving in the wrong direction. Regularity is only aviodance of what ZFC is heading at, i.e. Russells set. Regularity is like putting your hands in front of your eyes as not to see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the need help with, you will find a theory that is as consistent as ZFC and do not have problems with having a universe. Any set theory which has a paradoxial universe, is a set thoery of LIES. But it doesn't mean that it is not consistent, a theory that always lies is a consistent theory,as far as it avoids its universe, like ZFC, having Regularity as its braking power of avoidance of its universe. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the restrictions on the comprehension principle. Adding regularity provably makes no difference to the consistency of the theory, and the theory with regualarity can be interpreted in the theory without regularity and vice versa. > IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? > Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the > restrictions on the comprehension principle. Adding regularity provably > makes no difference to the consistency of the theory, and the theory > with regualarity can be interpreted in the theory without regularity > and vice versa. IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair I think there is a misunderstanding here between us as to the meaning of Regularity, to you Regularity means the axiom of regularity. To me, Regularity means a set that is not in itself. Because ZFC deals with sets that are not in themselfs, what I call them Regular sets, then Russell's paradox will be raised when the set of all sets in ZFC is put on the table. The Axiom of Regularity, serves to prevents any set in ZFC from being this Russell set., i.e. in ZFC you will never reach the set of all sets, IF I am to put it in other words, I can say that Russell set (i.e. the set of all ZFC sets) IS the LIMIT of ZFC. ( what I mean by the limit of ZFC, is that it is not a set in ZFC, but all ZFC sets are heading towards it but never reaching it.) Perhaps I am wrong regarding the axiom of regularity rule, u say that putting restrictions on comprehension serves that rule, Nevertheless weather it is regularity of restricted comprehension that makes Russell's set a limit set for ZFC, I don't care, since both are not a directional change in ZFC. A directional change in ZFC , happens if retheorize it, in such a manner that every set in ZFC is a set in itself. Axiom2). Here we will reach into a theory that have a universe that is in it ( Upper bound ). without any inconsistencies involoved. You say that 2) and 3) are incoherent. what do you mean exactly by thisincoherent. IF you mean not understandable they u are wrong since it is clear what the statement Ax (xex), it means that Every x -> x in x, in more lay language it means that Every set is a member of itself. 3) means that there exist only one set that is singlton and a member of itself. and this set is symolized as U. 4) the complementary set means, that there exist a complementary set for every set. A complementary set of y is a set which contain all x: x not in y, and do not contain any x:xey. or in other words if y' is the complementary set of y then y and y' are disjoint sets, i.e y.y'= {y.y'} ( in this theory A and B are called disjoint if A.B={A.B}=U.) ,and if y' contains all x:x not in y. so y'.y= {y'.y} <-> y and y' are disjoint. Now if 2)3) and 4) are not coherent to you, then I have just gave you their meaning, can you write what I have just said in a coherent formal logical way that every body can understand it. The axiom of infinity , is the one which I need help with, but appearantly nobody is want to supply it. it simpley means that if UeN then U'eN then {U,U'}'eN then {U,U',{U,U'}'} etc.... or more simpley it means if U e N , then the complement set of the union set of U, is in N. this would be U' ( since the union set of U is U itself ). Now since U' e N, then the complementary set of the union of U and U' is also in N. this would be {U,U'}'. Now since {U,U'}' is in N, then the complementary set of the union of U,U',{U,U'}' is also in N and that is { U,U',{U,U'}' }'. and since { U,U',{U,U'}' }' e N, then the complementary set of the union of U,U',{U,U'}',{ U,U',{U,U'}' }' is also in N, and that would be {U,U',{U,U'}',{ U,U',{U,U'}' }'}' And so on indefinitelly. Now how can I put all of this in a nice formal language. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the > restrictions on the comprehension principle. Adding regularity provably > makes no difference to the consistency of the theory, and the theory > with regualarity can be interpreted in the theory without regularity > and vice versa. IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair I think there is a misunderstanding here between us as to the meaning > of Regularity, to you Regularity means the axiom of regularity. To > me, Regularity means a set that is not in itself. Because ZFC deals > with sets that are not in themselfs, what I call them Regular sets, > then Russell's paradox will be raised when the set of all sets in ZFC > is put on the table. The Axiom of Regularity, serves to prevents any set in ZFC from being > this Russell set., Well, actually logic alone does that. Logic alone proves that the Russell set can't exist. > i.e. in ZFC you will never reach the set of all > sets, IF I am to put it in other words, I can say that Russell set > (i.e. the set of all ZFC sets) IS the LIMIT of ZFC. ( what I mean by > the limit of ZFC, is that it is not a set in ZFC, but all ZFC sets are > heading towards it but never reaching it.) Perhaps I am wrong regarding > the axiom of regularity rule, u say that putting restrictions on > comprehension serves that rule, Nevertheless weather it is regularity > of restricted comprehension that makes Russell's set a limit set for > ZFC, I don't care, since both are not a directional change in ZFC. A > directional change in ZFC , happens if retheorize it, in such a manner > that every set in ZFC is a set in itself. Axiom2). Here we will reach > into a theory that have a universe that is in it ( Upper bound ). > without any inconsistencies involoved. You say that 2) and 3) are incoherent. what do you mean exactly by > thisincoherent. > IF you mean not understandable they u are wrong since it is clear > what the statement > Ax (xex), it means that Every x -> x in x, in more lay language it > means that Every set is a member of itself. > (1) and (2) are fine. It was (3) that I found poorly expressed. > 3) means that there exist only one set that is singlton and a member of > itself. and this set is symolized as U. > So (3) just says there's only one singleton. Fine. > 4) the complementary set means, that there exist a complementary set > for every set. A complementary set of y is a set which contain all x: x not in y, and > do not contain any x:xey. or in other words if y' is the complementary > set of y then y and y' are disjoint sets, i.e y.y'= {y.y'} ( in this > theory A and B are called disjoint if A.B={A.B}=U.) ,and if y' contains > all x:x not in y. so y'.y= {y'.y} <-> y and y' are disjoint. Now if 2)3) and 4) are not coherent to you, then I have just gave you > their meaning, can you write what I have just said in a coherent formal > logical way that every body can understand it. > Yes, I understand them now. And I can prove that they are consistent, so that's good. > The axiom of infinity , is the one which I need help with, but > appearantly nobody is want to supply it. it simpley means that if UeN then U'eN then {U,U'}'eN then > {U,U',{U,U'}'} etc.... or more simpley it means if U e N , then the complement set of the > union set of U, is in N. > this would be U' ( since the union set of U is U itself ). Now since U' e N, then the complementary set of the union of U and U' > is also in N. > Hang on a moment. If U' is the complement of U, then the union of U and U', if it existed, would be a universal set - a set containing everything. But then its complement would have to be empty, and we can't have empty sets since every set is a member of itself. > this would be {U,U'}'. Now since {U,U'}' is in N, then the complementary set of the union of > U,U',{U,U'}' is also in N and that is { U,U',{U,U'}' }'. and since { U,U',{U,U'}' }' e N, then the complementary set of the > union of > U,U',{U,U'}',{ U,U',{U,U'}' }' is also in N, and that would be > {U,U',{U,U'}',{ U,U',{U,U'}' }'}' And so on indefinitelly. Now how can I put all of this in a nice formal language. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the > restrictions on the comprehension principle. Adding regularity provably > makes no difference to the consistency of the theory, and the theory > with regualarity can be interpreted in the theory without regularity > and vice versa. IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair I think there is a misunderstanding here between us as to the meaning > of Regularity, to you Regularity means the axiom of regularity. To > me, Regularity means a set that is not in itself. Because ZFC deals > with sets that are not in themselfs, what I call them Regular sets, > then Russell's paradox will be raised when the set of all sets in ZFC > is put on the table. The Axiom of Regularity, serves to prevents any set in ZFC from being > this Russell set., Well, actually logic alone does that. Logic alone proves that the > Russell set can't exist. Yes , that is true, but that is something else, we are talking about a set theory that is a LIE, or a consistent lie, the subject is not weather logic itself is a lie, of course logic is not a lie. i.e. in ZFC you will never reach the set of all > sets, IF I am to put it in other words, I can say that Russell set > (i.e. the set of all ZFC sets) IS the LIMIT of ZFC. ( what I mean by > the limit of ZFC, is that it is not a set in ZFC, but all ZFC sets are > heading towards it but never reaching it.) Perhaps I am wrong regarding > the axiom of regularity rule, u say that putting restrictions on > comprehension serves that rule, Nevertheless weather it is regularity > of restricted comprehension that makes Russell's set a limit set for > ZFC, I don't care, since both are not a directional change in ZFC. A > directional change in ZFC , happens if retheorize it, in such a manner > that every set in ZFC is a set in itself. Axiom2). Here we will reach > into a theory that have a universe that is in it ( Upper bound ). > without any inconsistencies involoved. You say that 2) and 3) are incoherent. what do you mean exactly by > thisincoherent. > IF you mean not understandable they u are wrong since it is clear > what the statement > Ax (xex), it means that Every x -> x in x, in more lay language it > means that Every set is a member of itself. > (1) and (2) are fine. It was (3) that I found poorly expressed. 3) means that there exist only one set that is singlton and a member of > itself. and this set is symolized as U. > So (3) just says there's only one singleton. Fine. 4) the complementary set means, that there exist a complementary set > for every set. A complementary set of y is a set which contain all x: x not in y, and > do not contain any x:xey. or in other words if y' is the complementary > set of y then y and y' are disjoint sets, i.e y.y'= {y.y'} ( in this > theory A and B are called disjoint if A.B={A.B}=U.) ,and if y' contains > all x:x not in y. so y'.y= {y'.y} <-> y and y' are disjoint. Now if 2)3) and 4) are not coherent to you, then I have just gave you > their meaning, can you write what I have just said in a coherent formal > logical way that every body can understand it. > Yes, I understand them now. And I can prove that they are consistent, > so that's good. The axiom of infinity , is the one which I need help with, but > appearantly nobody is want to supply it. it simpley means that if UeN then U'eN then {U,U'}'eN then > {U,U',{U,U'}'} etc.... or more simpley it means if U e N , then the complement set of the > union set of U, is in N. > this would be U' ( since the union set of U is U itself ). Now since U' e N, then the complementary set of the union of U and U' > is also in N. > Hang on a moment. If U' is the complement of U, then the union of U and > U', if it existed, Of course it exist. There is not problem with it in this theory. would be a universal set - a set containing > everything. But then its complement would have to be empty, and we > can't have empty sets since every set is a member of itself. Yes, this is true. I missed that. In reality I should correct 4) in such a manner as to exempt {} from being a set in this theory. the complementary set means, that there exist a complementary set for every set other than the set of all sets. Ax.4) Axiom of Completion: AyExAz z=/={ s|ses } zex <-> z!ey About Infinity I said I need help about it. It is not completed yet. I need one to help me frame an axiom that is comparable to the axiom of infinity in ZFC. In addtion I want to ask about the axiom of separation which exists in ZFC, were is the restriction exactly. The axiom is like this: AyExAz zex <-> zey / P(y) If P is any predicate, then were is the restriction of comprehension. Now what can an unrestricted for of comprehension be. or what is called Naive comprehension, what is exactly its formula. Is it :_ for Every P that is a predicate in one variable, there exist a set x, such that x|P(x) Ex x|P(x). Zuhair this would be {U,U'}'. Now since {U,U'}' is in N, then the complementary set of the union of > U,U',{U,U'}' is also in N and that is { U,U',{U,U'}' }'. and since { U,U',{U,U'}' }' e N, then the complementary set of the > union of > U,U',{U,U'}',{ U,U',{U,U'}' }' is also in N, and that would be > {U,U',{U,U'}',{ U,U',{U,U'}' }'}' And so on indefinitelly. Now how can I put all of this in a nice formal language. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the > restrictions on the comprehension principle. Adding regularity provably > makes no difference to the consistency of the theory, and the theory > with regualarity can be interpreted in the theory without regularity > and vice versa. IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair I think there is a misunderstanding here between us as to the meaning > of Regularity, to you Regularity means the axiom of regularity. To > me, Regularity means a set that is not in itself. Because ZFC deals > with sets that are not in themselfs, what I call them Regular sets, > then Russell's paradox will be raised when the set of all sets in ZFC > is put on the table. The Axiom of Regularity, serves to prevents any set in ZFC from being > this Russell set., Well, actually logic alone does that. Logic alone proves that the > Russell set can't exist. Yes , that is true, but that is something else, we are talking about a > set theory that is a LIE, > or a consistent lie, the subject is not weather logic itself is a lie, > of course logic is not a lie. i.e. in ZFC you will never reach the set of all > sets, IF I am to put it in other words, I can say that Russell set > (i.e. the set of all ZFC sets) IS the LIMIT of ZFC. ( what I mean by > the limit of ZFC, is that it is not a set in ZFC, but all ZFC sets are > heading towards it but never reaching it.) Perhaps I am wrong regarding > the axiom of regularity rule, u say that putting restrictions on > comprehension serves that rule, Nevertheless weather it is regularity > of restricted comprehension that makes Russell's set a limit set for > ZFC, I don't care, since both are not a directional change in ZFC. A > directional change in ZFC , happens if retheorize it, in such a manner > that every set in ZFC is a set in itself. Axiom2). Here we will reach > into a theory that have a universe that is in it ( Upper bound ). > without any inconsistencies involoved. You say that 2) and 3) are incoherent. what do you mean exactly by > thisincoherent. > IF you mean not understandable they u are wrong since it is clear > what the statement > Ax (xex), it means that Every x -> x in x, in more lay language it > means that Every set is a member of itself. > (1) and (2) are fine. It was (3) that I found poorly expressed. 3) means that there exist only one set that is singlton and a member of > itself. and this set is symolized as U. > So (3) just says there's only one singleton. Fine. 4) the complementary set means, that there exist a complementary set > for every set. A complementary set of y is a set which contain all x: x not in y, and > do not contain any x:xey. or in other words if y' is the complementary > set of y then y and y' are disjoint sets, i.e y.y'= {y.y'} ( in this > theory A and B are called disjoint if A.B={A.B}=U.) ,and if y' contains > all x:x not in y. so y'.y= {y'.y} <-> y and y' are disjoint. Now if 2)3) and 4) are not coherent to you, then I have just gave you > their meaning, can you write what I have just said in a coherent formal > logical way that every body can understand it. > Yes, I understand them now. And I can prove that they are consistent, > so that's good. The axiom of infinity , is the one which I need help with, but > appearantly nobody is want to supply it. it simpley means that if UeN then U'eN then {U,U'}'eN then > {U,U',{U,U'}'} etc.... or more simpley it means if U e N , then the complement set of the > union set of U, is in N. > this would be U' ( since the union set of U is U itself ). Now since U' e N, then the complementary set of the union of U and U' > is also in N. > Hang on a moment. If U' is the complement of U, then the union of U and > U', if it existed, Of course it exist. There is not problem with it in this theory. would be a universal set - a set containing > everything. But then its complement would have to be empty, and we > can't have empty sets since every set is a member of itself. Yes, this is true. I missed that. In reality I should correct 4) in > such a manner as to exempt {} from being a set in this theory. > the complementary set means, that there exist a complementary set > for every set other than the set of all sets. Ax.4) Axiom of Completion: AyExAz z=/={ s|ses } zex <-> z!ey About Infinity I said I need help about it. It is not completed yet. I need one to help me frame an axiom that is > comparable to the axiom of infinity in ZFC. In addtion I want to ask about the axiom of separation which exists in > ZFC, were is the restriction exactly. The axiom is like this: AyExAz zex <-> zey / P(y) If P is any predicate, then were is the restriction of comprehension. Now what can an unrestricted for of comprehension be. or what is called Naive comprehension, what is exactly its formula. Is it :_ for Every P that is a predicate in one variable, there exist a > set x, such that x|P(x) Ex x|P(x). Zuhair comprehension: Naive comprehension is the principle that, for every proposition P with one free variable x, there is a set {x | P(x)}. These are Jesse's words. I think I can symbolize this as: EyAxAP(x) y={x|P(x)} If I got it correct, should I add that as an axiom to the rest of axioms. since I don't need any restrictions on comprehension. since this set theory is not heading at a Russell set as ZFC does. I think if I add this axiom, then I will currenty that {x|xex} is a set in this theory. Without any inconsistency involoved ( unlike ZFC which needs restrictions for it to be in the safty zone of logical consistency ). I like to symbolize ZFC in the following manner: {}-ZFC, {x|x is a set according to ZFC} The dash- between ZFC and {} means that {} is a set in ZFC, and so {} is the minimum lower bound. while the comma , between ZFC and {x|x is a set according to ZFC} means that the later is not a set in ZFC, and it surves as The LIMIT of ZFC. so ZFC has no universe. While my correction of ZFC, symbolized as Z.ZFC is as follows: U-Z.ZFC-{x|x is a set according to Z.ZFC} You see Z.ZFC has both minimal lower bound and maximum upper bound. That's why I think it is more consistent than ZFC itself. We can communicate consistently with a man or a system that always lies, We also can communicate consistently with a man or a system that always tells the truth. But to differentiate between both conditions, the ALWAYS LIER do not have a universe for his statements or models. While The truth teller has. For philosophical reasons, ZFC should be corrected to Z.ZFC. Zuhair > this would be {U,U'}'. Now since {U,U'}' is in N, then the complementary set of the union of > U,U',{U,U'}' is also in N and that is { U,U',{U,U'}' }'. and since { U,U',{U,U'}' }' e N, then the complementary set of the > union of > U,U',{U,U'}',{ U,U',{U,U'}' }' is also in N, and that would be > {U,U',{U,U'}',{ U,U',{U,U'}' }'}' And so on indefinitelly. Now how can I put all of this in a nice formal language. Zuhair === Subject: Re: Why Regularity? comprehension: Naive comprehension is the principle that, for every proposition P with one free variable x, there is a set {x | P(x)}. These are Jesse's words. I think I can symbolize this as: EyAxAP(x) y={x|P(x)} If I got it correct, should I add that as an axiom to the rest of axioms. since I don't need any restrictions on comprehension. since this set theory is not heading at a Russell set as ZFC does. I think if I add this axiom, then I will currenty that {x|xex} is a set in this theory. Without any inconsistency involoved ( unlike ZFC which needs restrictions for it to be in the safty zone of logical consistency ). I like to symbolize ZFC in the following manner: {}-ZFC, {x|x is a set according to ZFC} The dash- between ZFC and {} means that {} is a set in ZFC, and so {} is the minimum lower bound. while the comma , between ZFC and {x|x is a set according to ZFC} means that the later is not a set in ZFC, and it surves as The LIMIT of ZFC. so ZFC has no universe. While my correction of ZFC, symbolized as Z.ZFC is as follows: U-Z.ZFC-{x|x is a set according to Z.ZFC} You see Z.ZFC has both minimal lower bound and maximum upper bound. That's why I think it is more consistent than ZFC itself. We can communicate consistently with a man or a system that always lies, We also can communicate consistently with a man or a system that always tells the truth. But to differentiate between both conditions, the ALWAYS LIER do not have a universe for his statements or models. While The truth teller has. For philosophical reasons, ZFC should be corrected to Z.ZFC. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. I am speaking in a more philosophical manner, I see ZF as a theory of > finding consistency of statements of the that who always lies. All sets > in ZFC are lies. Yet , I said such M that all of its statements are > lies do have consistency, it is the consistency of the total lier, > though I call the later ( the opposite truth teller ) anyhow. This is > philosophical. Putting the axioms of regularity confines us to the lies > that are fairly consistent, it makes us avoid the lier paradox, which > reveal the philosophical truth to ZF. ZFC with Regularity is consistent > logically, but philosophically speaking, it is a system of consistent > lies, see the ulternative that I have proposed, I think this is the > HONEST set theory . But It seems not so practical. The Lieying ZF ( the > standard one ) is easier to deal with. And since all what we require > from a set theory is consistency, then it doesn't really matter if ZF > is philosophically a consistent lie. Since it easier to deal with, then > let it. Zuhair There is a mathematical analysis of the liar paradox. Tarski proved > that no language can define its own truth predicate. That's how the > liar paradox is avoided in mathematics. As I say, I don't see what the liar paradox has to do with the axiom of > regularity. You are not discriminating between , philosophy and mathematics. Yes I am. Read > about regularity only, it is about the whole of ZFC, the theorym of > consistent lies. Zuhair I don't think you're doing philosophy, I think you're doing meaningless > gibberish. I don't know why you like using these meaningless words. If you read > something you have to think about it, not just refuse it > straightforwards. There is great similarity between Russells sets and the Lier paradox, > you should be blind not to see this. Regularity aims at avoiding this paradox. If we use unrestricted > comprehension on ZF, and remove Regularity, then you will see ZF going > to its true nature. Just putting a restriction like regularity, though > avoids this paradox, but doens't let you reveal the truth of this > theory. It's not really regularity which avoids the paradox. It's the > restrictions on the comprehension principle. Adding regularity provably > makes no difference to the consistency of the theory, and the theory > with regualarity can be interpreted in the theory without regularity > and vice versa. IF you are driving your car in the wrong direction, and you discovered > that, and reduced your spead so that you will not reach the wrong aim, > it doesn't mean that you are not driving in the wrong direction. > Regularity is only aviodance of what ZFC is heading at, i.e. Russells > set. > Regularity is like putting your hands in front of your eyes as not to > see the doom your are heading into, it doesn't change anything. The direction should be changed. If you remove regularity and make some directional modefication on the > need help with, you will find a theory that is as consistent as ZFC and > do not have problems with having a universe. > Where have you defined this theory again? I had a look at your earlier posts. Your axioms 3 and 4 seem a bit incoherent. Any set theory which has a paradoxial universe, is a set thoery of > LIES. But it doesn't mean that it is not consistent, a theory that > always lies is a consistent theory,as far as it avoids its universe, > like ZFC, having Regularity as its braking power of avoidance of its > universe. Zuhair === Subject: Re: Why Regularity? > The reason why we have the axiom of regularity isn't really anything to > do with the liar paradox. It's just an indication of the fact that we > only want to study well-founded sets. In ZF-regularity we can prove > that the class of well-founded sets is a model for ZF with regularity > anyway. Can you help me with stating the axiom of infinity, ( see Thread ). Zuhair === Subject: Re: Why Regularity? Oops, I made a mistake. The Theory goes like the following: - Converse ZFC Set Theory- Primitive e 1) Axiom of Extentiality. 2) Axiom of Irregularity: Ax (xex) 3) Axiom of Uniqueness: EU,Ay,Az (zey <-> z=y) -> y=U There exist a unique set U that for every set y, every set z, zey <-> z=y , then y=U. 4) Axiom of Completion: AyExAz ( zex <-> z!ey ) For every set y , there exist a set x, such that for every set z that is in x, then z is not in y, and every z not in x, then z in y. x is called the complementary set of y. 5) Axiom of Pairing. 6) Axiom of Union. 7) Axiom of Separation. 8) Axiom of Replacement. 9) Axiom of Power set. 10) Axiom of Choice. All axioms except 2) , 3) , 4) , are as in ZFC. === Subject: Re: Why Regularity? > Oops, I made a mistake. The Theory goes like the following: - Converse ZFC Set Theory- > Primitive e 1) Axiom of Extentiality. > 2) Axiom of Irregularity: Ax (xex) > 3) Axiom of Uniqueness: EU,Ay,Az (zey <-> z=y) -> y=U > There exist a unique set U that for every set y, every set z, > zey <-> z=y , then y=U. 4) Axiom of Completion: AyExAz ( zex <-> z!ey ) > For every set y , there exist a set x, such that for every set z > that is in x, > then z is not in y, and every z not in x, then z in y. > x is called the complementary set of y. 5) Axiom of Pairing. > 6) Axiom of Union. > 7) Axiom of Separation. > 8) Axiom of Replacement. > 9) Axiom of Power set. > 10) Axiom of Choice. > All axioms except 2) , 3) , 4) , are as in ZFC. Another axiom can be added 11) Axiom of Infinity. N can be built in this way. first member of N is U, then by axiom 4) , there exist U' . Now by axiom 5) {U,U'} exist. by axiom 5 { U,{U,U'} } exist by axiom 5 { U,{U,U'},{ U,{U,U'} } } by axiom 5 { U,{U,U'},{ U,{U,U'} }, { U,{U,U'},{ U,{U,U'} } } } and so on I need help with formulatng this logically. Zuhair === Subject: Re: Why Regularity? > Another axiom can be added 11) Axiom of Infinity. N can be built in this way. first member of N is U, then by axiom 4) , there exist U' . Now by axiom 5) {U,U'} exist. by axiom 5) { U,{U,U'} } exist by axiom 5 { U,{U,U'},{ U,{U,U'} } } HELP!!!! I got this wrong. -------------------------------------------------------------------------- Let me start it again. Axiom of Infinity. Let U be a member of N subset of N = {U} Now by axiom 4 , U' exist and let it be in N. subset of N = { U,U'} by axiom 4, { U,U'}' exist, and Let it be in N. subset of N = { U,U', {U,U'}' } by axioms4) { U,U', {U,U'}' }' exist and let it be in N subset of N= { U, U',{U,U'}' , { U,U', {U,U'}' }' } by axiom 4) { U, U',{U,U'}' , { U,U', {U,U'}' }' }' exist and let it be in N subset of N = { U, U',{U,U'}' , { U,U', {U,U'}' }' ,{ U, U',{U,U'}' , { U,U', {U,U'}' }' }' } and so on, infinitelly........... Now N= Union of All subsets of N. This N is similar to the set of all natural numbers present in ZFC. we can say that U=0 , U'=1, {U,U'}' =2 , { U,U', {U,U'}' }' =3 , { U, U',{U,U'}' , { U,U', {U,U'}' }' }' = 4, .....,etc. I need HELP to formulate this axiom logically? Zuhair === Subject: Re: Are there functions, such that...? >Are there functions f1(x) f2(x) f3(x), such that >d/dx f1 = C1 d/dx f2 >d/dx f2 = C2 d/dx f3 >d/dx f3 = C3 d/dx f1 >where C1,C2,C3 are constants, f1,f2,f3 are continuous. f1(x) = f2(x) = f3(x) = e^x c1 = c2 = c3 = 1. -- === Subject: Re: Are there functions, such that...? Are there functions f1(x) f2(x) f3(x), such that >d/dx f1 = C1 d/dx f2 >d/dx f2 = C2 d/dx f3 >d/dx f3 = C3 d/dx f1 >where C1,C2,C3 are constants, f1,f2,f3 are continuous. f1(x) = f2(x) = f3(x) = e^x > c1 = c2 = c3 = 1. but that is not most general take |m C_m x | e |3 which is (m,3)-multisection of the exponential for m={0, 1, 2} these functions obey the given equations and provide a basis to all such solutions they are a part of my generalised trigonometry that i have shown on these groups -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: Are there functions, such that...? >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous. f1(x) = f2(x) = f3(x) = e^x > c1 = c2 = c3 = 1. -- f1=f2=f3= any continuous function C1=C2=C3= 1 === Subject: Re: Are there functions, such that...? >Are there functions f1(x) f2(x) f3(x), such that >>d/dx f1 = C1 d/dx f2 >>d/dx f2 = C2 d/dx f3 >>d/dx f3 = C3 d/dx f1 >>where C1,C2,C3 are constants, f1,f2,f3 are continuous. f1(x) = f2(x) = f3(x) = e^x > c1 = c2 = c3 = 1. -- f1=f2=f3= any continuous function Differential. > C1=C2=C3= 1 --- === Subject: Re: Are there functions, such that...? Are there functions f1(x) f2(x) f3(x), such that >d/dx f1 = C1 d/dx f2 >d/dx f2 = C2 d/dx f3 >d/dx f3 = C3 d/dx f1 >where C1,C2,C3 are constants, f1,f2,f3 are continuous. >> f1(x) = f2(x) = f3(x) = e^x >> c1 = c2 = c3 = 1. >> -- >> f1=f2=f3= any continuous function Differential. > C1=C2=C3= 1 --- > Right. I saw too late that this thread was repeated and this was already covered. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= <455E5586.3090100@uchicago.edu> simple, because ?F isn?t an absoluta scale. alen Isenhart u#440483 my game: http://uc1.gamestotal.com/?tft=de53 === Subject: probability question A person give me a probability question wich i can't resolve A set of numbers N is given and a subset SN of that set N. Then I take random each time another subset SSN of N and I delete each time the numbers in SN wich are in SSN. What is the mean time to take SSN for all the numbers of SN are deleted? f.e. N = 1,2,3,4,5,6 SN = 1,2,3,4 SSN = two numbers SSN1 = 1,5 (SN = 2,3,4) SSN2 = 1,2 (SN = 3,4) SSN3 = 3,4 (SN empty) Here, I have taken three times a SNN. With the computer I have calculated with the Monte Carlo method and became 22.6..... for N = 42 numbers , Sn = 10 numbers from N and SNN= always 6 numbers from N Hope 22.6... is correct but I have no formula for the probability Any help? === Subject: Re: probability question > A person give me a probability question wich i can't resolve > A set of numbers N is given and a subset SN of that set N. You are getting bogged down in notation. Use S instead of SN; it's clear that S is a subset of N from the context. > Then I take random each time another subset SSN of N and I delete each time > the numbers in SN wich are in SSN. Is the new subset (call it T, not SSN) chosen uniformly? That is, are the probabilities of choosing every subset the same? If so, then the elements of S act independently, and each number is expected to last for 1 + 1/2 + 1/4 + 1/8 + ... 2 iterations. > What is the mean time to take SSN for all the numbers of SN are deleted? f.e. > N = 1,2,3,4,5,6 > SN = 1,2,3,4 > SSN = two numbers > SSN1 = 1,5 (SN = 2,3,4) > SSN2 = 1,2 (SN = 3,4) > SSN3 = 3,4 (SN empty) > Here, I have taken three times a SNN. Every T (SSN) here has 2 elements in it, which suggests that the subsets are NOT chosen uniformly at random. This alters the problem substantially. --- > With the computer I have calculated with the Monte Carlo method and became > 22.6..... > for N = 42 numbers , Sn = 10 numbers from N and SNN= always 6 numbers from > N > Hope 22.6... is correct but I have no formula for the probability > Any help? === Subject: Re: probability question In separate posts to sci.math and sci.stat.math, Andreas (andreas@pandora.be) posed the following problem, I think. I clean up the notation. Let k be a nonnegative number and m and n be positive integers with maj(m,k) <= n. Let A0 = {1, 2, 3,..., k}. For each positive integer i, let B_i be a random subset of size m from {1, 2, 3,..., n}; the B_i are independent. For nonnegative i, let A_(i+1) = A_i B_(i+1) (set difference). We are interested in R = min{i: A_i = {}}. [Andreas originally specified that m <= k, an unnecessary restriction.] Here is a recursive way to determine the expected value of R. We will keep m and n fixed and let r_k = ER for a given k. We will condition on S = the number of elements in the intersection of A0 and B1. Note that S has hypergeometric distribution: P{S = j} = C(k,j) C(n-k, m-j) / C(n,m) for maj(m+k-n, 0) <= j <= min(k, m), where C(a, b) denotes the binomial coefficient, a choose b. r0 = 0, and r_k = ER = EE[R | S] = 1 + sum(j= maj(m+k-n, 0).. min(k, m), P{S = j} r_(k-j)) for k > 0. Note that if 0 < k <= m, this becomes r_k = [1 + sum(j= maj(m+k-n, 0).. k-1, P{S = j} r_(k-j))] / (1 - P{S = k}). Similarly, one can derive a recursive formula for v_k = Var(R) via the identity Var(R) = EVar(R | S) + Var(E[R | S]). In fact, one can similarly derive the probability generating function for R. In all cases, I would rely on a CAS system such as Mathematica or Maple. I am contemplating another approach. Something along the lines of letting R_j = min{i : j not in A_i} for j = 1, 2,..., k. Then R = min(R1, R2,..., R_k). A priori, it is not clear that this will work. Note that when m = 1, R is simply a sum of independent geometrics with decreasing parameters. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: probability question What's in a name, Proginoskes? I think one can see what I mean in my given example. I take N numbers and I take one subset SN from N. Then I take several subsets SSN (always k numbers random from the original N) and delete in SN the numbers wich are found in SSN. How many subsets SSN had I normaly to take till SN is empty? 22.6... Proginoskes schreef in bericht > A person give me a probability question wich i can't resolve >> A set of numbers N is given and a subset SN of that set N. You are getting bogged down in notation. Use S instead of SN; it's > clear that S is a subset of N from the context. > Then I take random each time another subset SSN of N and I delete each >> time >> the numbers in SN wich are in SSN. Is the new subset (call it T, not SSN) chosen uniformly? That is, are > the probabilities of choosing every subset the same? If so, then the elements of S act independently, and each number is > expected to last for > 1 + 1/2 + 1/4 + 1/8 + ... 2 iterations. > What is the mean time to take SSN for all the numbers of SN are deleted? >> f.e. >> N = 1,2,3,4,5,6 >> SN = 1,2,3,4 >> SSN = two numbers >> SSN1 = 1,5 (SN = 2,3,4) >> SSN2 = 1,2 (SN = 3,4) >> SSN3 = 3,4 (SN empty) >> Here, I have taken three times a SNN. Every T (SSN) here has 2 elements in it, which suggests that the > subsets are NOT chosen uniformly at random. This alters the problem > substantially. --- > With the computer I have calculated with the Monte Carlo method and >> became >> 22.6..... >> for N = 42 numbers , Sn = 10 numbers from N and SNN= always 6 numbers >> from >> N >> Hope 22.6... is correct but I have no formula for the probability >> Any help? > === Subject: Re: simplifying brackets > In a high school text I saw the question: simplify (x)^2. The answer > in the back of the book was x^2. But how can this be true for negative > bases, because I thought that 3^2 =/= -3^2, (exponents taking > precedence over negation). The - is already part of a negative number. > If so, how can (x)^n = x^n for x<0 and even > n? What am I missing here? Typographical manipulation of equations does not preserve correctness. For instance, 2 + 2 = 4, but if I glue a 3 onto the end of both sides of that equation, I get 2 + 23 = 43, which is false. --- === Subject: Re: simplifying brackets > In a high school text I saw the question: simplify (x)^2. The answer > in the back of the book was x^2. But how can this be true for negative > bases, because I thought that 3^2 =/= -3^2, (exponents taking > precedence over negation). The - is already part of a negative number. But which negative number? does -3^2 mean (-3)^2 or -(3^2) ? I side with -(3^2), as having negation on a par with subtraction. it would be a bit awkward to have -3^2 > 0-3^2. If so, how can (x)^n = x^n for x<0 and even > n? What am I missing here? Typographical manipulation of equations does not preserve correctness. For instance, 2 + 2 = 4, but if I glue a 3 onto the end of both sides of that equation, I get 2 + 23 = 43, which is false. --- === Subject: Re: simplifying brackets In a high school text I saw the question: simplify (x)^2. The answer > in the back of the book was x^2. But how can this be true for negative > bases, because I thought that 3^2 =/= -3^2, (exponents taking > precedence over negation). The - is already part of a negative number. But which negative number? does -3^2 mean (-3)^2 or -(3^2) ? The original question is really: If (x)^2 = x^2, then what happens if x = -3? If so, then the - is part of the x, In the expression -3^2, the - is not part of the 3, though. --- > I side with -(3^2), as having negation on a par with subtraction. it would be a bit awkward to have -3^2 > 0-3^2. If so, how can (x)^n = x^n for x<0 and even > n? What am I missing here? Typographical manipulation of equations does not preserve correctness. For instance, 2 + 2 = 4, but if I glue a 3 onto the end of both sides of that equation, I get 2 + 23 = 43, which is false. --- === Subject: Re: JSH: Not a Mensa fan That ended up being more frating, though at least one group > published my paper in their book of mostly, um, puzzles. [jsh_is_a_tard@yahoo.com] >> I puzzled ur moms ass last night [T. Ashley] > I haven't watched sci.math for quite a while. Things have heated > up, I see. > We now have [at least] one individual who has assumed an identity to > attack> [at least] one other individual. jsh_is_a_tard now claims to be having romantic liaisons with > JSH's mom. I am absolutely speechless. [Proginoskes] >> It gets worse ... jsh_is_a_tard and JSH are the same person. [T. Ashley] > Is this conjecture or is there strong evidence? Choice #3: Proginoskes was joking, as part of long-running ridicule > centered on JSH's long-running obsession with posters' Secret Identities. Bingo. Now, for the bonus question: Can you find the insult inherent in the assumption that jsh_is_a_tard and JSH are the same person? (Hint: All of the necessary information is contained in this post and the quoted posts.) --- === Subject: Re: JSH: Not a Mensa fan [Tim Peters] >> ... >> and would bet 25 cents JSH never posts as anyone other than himself [sg552@hotmail.co.uk] > I don't think he does regularly, though I was rather suspicious > recently. I wondered briefly about the same msg at first, but it didn't pass my JSH sniff-test; for example, I don't think premier is in his working vocabulary. > On the 8th of November, in this post: > Going forward you may not know where the next move will be. You may >> not know--until it's too late--what person I know who is on my side, >> or in my corner, and you will not know the consequences for your >> life, until those consequences are a reality. >> Open information and open source were cornerstones of my >> philosophy--but no longer. >> The math wars have convinced me that some ideals are meant to be >> lost, and that letting your opponents know too much about what is >> coming is just a bad idea. Well, he routinely announces a fundamental shift in strategy that will leave his imagined enemies helpless. Later in the same message, he clarified that the people on his side they won't know about before it's too late are the most powerful people on the planet, and in closing gave the US-centric example that he would remove your defenses before you even know that some senator or governor is breathing down your necks. Following through on such grandiose threats with a sock puppet asking about > And then, the following day, a poster who goes by Ernest Praline, who > this topic: > Yup, a message asking: has keeled over and died. It just disappeared into thin air. Anyone know what happened to it? > Looked kind of suspicious to me. James didn't post for a few days, > and Ernest disappeared soon after. Perhaps I'm just paranoid though. You are (IMO), but it's contagious :-) Ernest Praline didn't post to that thread again either, but did start the goofy-fun Conjecture on thread length thread. In fact, you confronted him about the possibility he was JSH in that, and his reply erased all doubt in my mind: [ernest_praline@hotmail.com] > Conjecture: this thread will exceed 100 posts. [sg552@hotmail.co.uk] >> You are still James Harris, and I still claim my five pounds. [ernest_praline@hotmail.com] > There's mounting evidence that may conjecture will prove to > be true. When did that ever happen to James Harris? That was a funny reply! Unlikely to be James for that reason alone. Also shows that Praline is well aware of Harris, suggesting he's a jokester just there's now mounting evidence that his conjecture won't prove to be true after all :-) === Subject: Re: An infinite debate <4552D94E.5050504@et.uni-magdeburg.de> <45542EDF.70007@et.uni-magdeburg.de> <4558574A.5020604@et.uni-magdeburg.de Eckard Blumschein ha escrito: > A first imperfection is obvious lack of any solid basis: >> Dedekind himself admitted that the basis of his cuts cannot be proven. >> Cantor's interpretation of 2nd DA ignores 4th logic possibility. >> Zermelo's evidence of well-ordering is based on exhaustion of infinity. >> Cantor's definition of transfinite set has been confessed invalid and >> impossible to correct without abandoning the belief basic to set theory. >> Alleged refutation of insights by Aristotele, Spinoza, Gauss and others >> turns out to be pure rethoric by Cantor without substance. >> A second and equally important class of imperfections relates to >> application. There are and I gave so many examples that I do not have >> the time to recall them all just in order to be polite towards Vergil. >> Eckard Blumschein > **************************************************************************** > I wonder whether you're a succesful student of Mueckheim (sp?) in that > rather suspicious college in Germany (perhaps in ex-East Germany...?) > or whatever that has him as a teacher... Prof. Mueckenheim has been teaching in Augsburg near to Munich, Bavaria. > I never met him. Magdeburg is located in the central part of Germany, > near to Halle where Cantor lived and Braunschweig where Dedekind lived. I am not a mathematician, just someone who is used to think consequently > and who got aware of imperfections when I applied mathematics. ********************************************************************* Hi: == Obviously you're not a mathematician, that was unnecessary to stress. == About you being used to think consequently: your posts attacking Dedekind and Cantor (this is funny...), and mathematicians that accept this theory (this is also funny, but at least living mathematicians can respond...) say otherwise. == And about being aware of imperfections when you applied mathematics (could you give 2-3 examples? For the fun of it...): has it ever occured to you that it could be that YOU are too lousy an engineer (or whatever) and that you don't understand basic maths well enough to see that you're doing dumb things? If I intend to fix my car and pour some water with sugar into the fuel tank and then my car doesn't work, accusing the car manufacturer and/or the one that invented engines would be rather idiotic, don't you think? How do you, with your rather limited and deformed mathematical education and your ackward, to say the least, logic, know for sure that you are right in a mathematical debate and ALL mathematicians are wrong? Well, that's the mark of the crank, I believe: ALL are wrong, I Your continuously making references, some of which are at least extremely moot and, perhaps, false, to words said by great men in the past don't REALLY support your rather weak and pitiful arguments...you are aware of this, right? But in the meantime we write a lot in english... Tonio > You apparently reject the strong, unreffuted-so-far, I collected enough arguments as to completely refute it. Let's resume > the battle seemingly won by the believer in what I consider an Utopia. Set Theory > development by Dedekind, Cantor AND MANY OTHERS (Hey! Frege, Russell, > Hilbert, Zorn, etc. were also there!) because it does not abide to ANY > of your rather limited physical, real or whatever views. Do not confuse me with Mueckenheim, Cantor or Hilbert who in 1925 also > looked at physics. > My view leans on a purely logical basis without any regard to physics. Ok, but then what basis do you have to ask, and even insult, formed and > serious mathematicians because they do so? Logic. Only because you can't buy > aleph_null balls or because you don't know any infinite hotel Hibert's you want > to disregard a beautiful, CONSISTENT, and useful mathematical theory? I understand your feelings. Any Utopie is consistent inside its own > paradise. What about usefulness, I wonder if someone has any use for > aleph_2. Aleph_0 rougly means infinite, aleph_1 roughly corresponds to > uncountable. > Ha...! Good luck in that, I'm sure your efforts are doomed to > failure... I am working hard on making the basic failure by Dedekind and Cantor > obvious even to you. As soon as science is independent of belief, every > defense of unfounded basics is doomed to fail. Eckard === Subject: Re: integration over a levelset surface > hi, can anybody help me guide how to integrate over a levelset surface. > suppose I have a function f: R^n X [0,1] -->R for which zero lecelset > is > S(t)={x: f(x,t)=0} Now, i have another function g: R^n ->[0,1] defined over R^n. i want to integrate g(x) over S(t). What is your definition of int_{S(t)} g ? That might be helpful to give hints how to solve the problem. > can anybody help how to proceed to evaluate/estimate this integral? > J. === Subject: Re: integration over a levelset surface <455EC018.50305@web.de> well think of g1(x) and g2(x) as probablity densities for two different classes. let t denotes the prior probablity for class having density g1, and (1-t) the prior probablity of class having density g1. so this forms a finite mixture model with marginal density g(x)=t*g1(x)+(1-t)*g2(x) now S(t)={x: t*g1(x)-(1-t)*g2(x)=0} is the bayesian boundary surface where these two class densities intersect. i'm trying to integrate g2(x) over S(t) > hi, can anybody help me guide how to integrate over a levelset surface. > suppose I have a function f: R^n X [0,1] -->R for which zero lecelset > is > S(t)={x: f(x,t)=0} Now, i have another function g: R^n ->[0,1] defined over R^n. i want to integrate g(x) over S(t). What is your definition of int_{S(t)} g ? That might be helpful to give > hints how to solve the problem. can anybody help how to proceed to evaluate/estimate this integral? > J. === Subject: Re: integration over a levelset surface <455EC018.50305@web.de> oooops!! (1-t) the prior probablity of class having density g2 === Subject: Re: a^8+b^8+c^8+d^8 = e^8+f^8+g^8+h^8 > Hi Tito, do you known a cubic polynomial which has roots that allow to generate > the famous taxicab number relationship 1729 or 1^3+12^3 = 9^3+10^3 ? > The roots of a cubic always allow to define a perfect 3rd power as > follows: > (6ax+2b)^3=72ab^2x+ 8b^3 - 6^3a^2(cx+d) > I was hoping that there exists a cubic which could generate all 4 terms > as cubes. > Gerry I see what you mean, (6ax+2b)^3-(72ab^2x+ 8b^3 - 6^3a^2(cx+d)) = 6^3a^2(ax^3+bx^2+cx+d) = 0 though for a,b, and (6ax+2b) rational, that implies x is also rational and your cubic is reducible. - === Subject: Re: Generalization of Monthly problem 11245 > abcmax := 100; > for c from 1 to abcmax do # 1 <= a <= b <= c <= abcmax > for b from 1 to c do > for a from 1 to b do > if (c <= a + b) then > rRratio := (a + b - c)*(a + c - b)*(b + c - a)/(2*a*b*c); > # Formula for r/R; > rside_squared := > 2 * (2*a^2 - (b - c)^2) > * (2*b^2 - (a - a)^2) (a-a)^2 - eeep! Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We T, Inc./. === Subject: Re: Generalization of Monthly problem 11245 > # American Mathematical Monthly problem 11245 > # (October, 2006 issue, p. 760 -- solution due Feb 28, 2007) says deadline. It defeats their purpose. The same applies to their generalizations. --- === Subject: independant familly of polynomial Hi all, I have a problem with this simple exercice. Suppose that P is a real polynomial of degree n, and a_0, a_1, a_2, .., a_n are distincts real numbers. How can one prove that P(x-a_0), P(x-a_1),.., P(x_a_n) is an independant familly of polynomials ? thankx, Fedor === Subject: Re: independant familly of polynomial >Hi all, I have a problem with this simple exercice. Suppose that P is a real >polynomial of degree n, and a_0, a_1, a_2, .., a_n are distincts real >numbers. How can one prove that P(x-a_0), P(x-a_1),.., P(x_a_n) is an >independant familly of polynomials ? thankx, >Fedor > Start with the basic case: if P(x)=x^n and c0 P(x-a0) + ... + cn P(x-an) = 0, then equating like powers of x leads to the system: V(a0,..,an) [c0 ... cn]^T = 0 where V is the Vandermonde matrix on a0,...,an. Since all of the a's are distinct, the determinant of V is not zero, and hence c0 = ... = cn = 0. In the general case, where P(x)=x^n + e_{n-1}x^{n-1} + ... + e0, you can proceed by induction to get the same system for the c's as in the basic case: The equation based on the coefficient of x^n is the same as when P(x)=x^n. If you substitute the equation for x^n into the equation for x^{n-1}, you get the same equation for x^{n-1} as when P(x)=x^n. In general, substituting the reduced equations for the coefficients of x^n, x^{n-1},... x^{k+1} into the equation for the coefficients of x^k leads to the same equation for x^k as when P(x)=x^n. -- Daniel Mayost === Subject: Re: Z.Ulternative Set Theory At last I have finished from building this set theory. - Ulternative Set Theory- > Primitive: e > 6) Separation Ax:10) Axiom of Universe > ExAy yex . > Is their anyone who think that this theory is inconsistent? > Yes.6) and 10) are incompatible using the standard Russell argument - Let U be a universe .(Ax 10) so Ax(xeU) > 6) if I understand this instance separation properly : AyEx (zex > <--->zey & z!ez) .Take y=U then using 10) (ie xeU always holds) we get : Ex (zex<-->zeU & z!ez) > <--> Ex(zex<-->z!ez) Let R (for Russell) be such an x .Then zeR <---> z!ez .Then as usual > take z=R to get ReR <--->R!eR which is a contradiction. Yes, I know this very well. But the contradiction is in membership of > U, it is not in the theorums which will spring from this theory. My > thinking is that contrdictive sets do exist as sets, they pose no > problem whatsoever on theoratic processing of this theory. But you are right regarding dropping the axiom of separation, since > there is the axiom of replacement, then there is not need for > separation. Either drop axiom 10) or call > your objects (what your variables denote (x,y,z,....) classes and > define x is a set iff Ey (xey) and then revise 10) to 10)' : Ex Ay (yex <--> y is a set) and call this x by U =x (the > universe of sets) .This is the Kelley -Moore system which is stronger > then Zermelo -Fraenkel .The Rest having to do with allowing xex and > friends is a denying of the regularity axiom which you are friee to do In reality I have tried something like this in another thread, but yet > I will try to put it here. So the theory would appear like that: > -Z.Ulternative Set Theory- Primitive e > Definition: x is a set <-> Ey (xey) Axioms: > 1) Extentiality 2) Universe: Ex Ay ( yex <-> y is a set ) 3) Empty set > 4) Circularity 5) Pairing6) Complementary set 7) Irregularity > 8) Ambiguity 9)Union 10) Infinity 11) Intersection 12) Harmony > 13) Synchrony 14) Replacement 15) Power 16) Choice. were Irregularity is what I called the axiom of irregular complementary > set. Zuhair Fine,now lets pretend we have recovered separation from replacement.Lets also name the universe (unique by extensionality ) U Let R be the class (objects denoted by variables) obtained from separation by requiring xeR <--->xeU and x!ex <--->xis a set and x!ex. follows that R is not a set. Non sets are called proper classes.Since R is a subclass of U it follows from the power set axiom that U is not a set either. However Ax( x is a subclass of U) .So all your objects are subclasses of the universe although some are not members of U. .You probably know all this but I wanted to check. This gets rid of contradictive sets.I strongly disagree that they cause no difficulty.Its true that in real life (serious ) reasoning using ordinary language there are contradictions that one avoids by watching out that you are making sense or using empirical checks using the senses ( eyes ,ears..) but in mathematics with its much limited scope but where the arguments get much more complicated it is intolerable to === Subject: Re: Z.Ulternative Set Theory At last I have finished from building this set theory. - Ulternative Set Theory- > Primitive: e > 6) Separation Ax:10) Axiom of Universe > ExAy yex . > Is their anyone who think that this theory is inconsistent? > Yes.6) and 10) are incompatible using the standard Russell argument - Let U be a universe .(Ax 10) so Ax(xeU) > 6) if I understand this instance separation properly : AyEx (zex > <--->zey & z!ez) .Take y=U then using 10) (ie xeU always holds) we get : Ex (zex<-->zeU & z!ez) > <--> Ex(zex<-->z!ez) Let R (for Russell) be such an x .Then zeR <---> z!ez .Then as usual > take z=R to get ReR <--->R!eR which is a contradiction. Yes, I know this very well. But the contradiction is in membership of > U, it is not in the theorums which will spring from this theory. My > thinking is that contrdictive sets do exist as sets, they pose no > problem whatsoever on theoratic processing of this theory. But you are right regarding dropping the axiom of separation, since > there is the axiom of replacement, then there is not need for > separation. Either drop axiom 10) or call > your objects (what your variables denote (x,y,z,....) classes and > define x is a set iff Ey (xey) and then revise 10) to 10)' : Ex Ay (yex <--> y is a set) and call this x by U =x (the > universe of sets) .This is the Kelley -Moore system which is stronger > then Zermelo -Fraenkel .The Rest having to do with allowing xex and > friends is a denying of the regularity axiom which you are friee to do In reality I have tried something like this in another thread, but yet > I will try to put it here. So the theory would appear like that: > -Z.Ulternative Set Theory- Primitive e > Definition: x is a set <-> Ey (xey) Axioms: > 1) Extentiality 2) Universe: Ex Ay ( yex <-> y is a set ) 3) Empty set > 4) Circularity 5) Pairing6) Complementary set 7) Irregularity > 8) Ambiguity 9)Union 10) Infinity 11) Intersection 12) Harmony > 13) Synchrony 14) Replacement 15) Power 16) Choice. were Irregularity is what I called the axiom of irregular complementary > set. Zuhair Fine,now lets pretend we have recovered separation from > replacement.Lets also name the universe (unique by extensionality ) U > Let R be the class (objects denoted by variables) obtained from > separation by requiring xeR <--->xeU and x!ex <--->xis a set and x!ex. follows that R is not a set. Non sets are called proper classes.Since R > is a subclass of U it follows from the power set axiom that U is not a > set either. However Ax( x is a subclass of U) .So all your objects are > subclasses of the universe although some are not members of U. .You probably know all this but I wanted to check. Yea, I know this of course. > This gets rid of contradictive sets.I strongly disagree that they cause > no difficulty.Its true that in real life (serious ) reasoning using > ordinary language there are contradictions that one avoids by watching > out that you are making sense or using empirical checks using the > senses ( eyes ,ears..) but in mathematics with its much limited scope > but where the arguments get much more complicated it is intolerable to Accordingly, this theory is consistent! Zuhair === Subject: irreducible polynomial of degree 5. Let p be a polynomial of degree 5 with coefficients in Q. If polynomials of degree 1 and 2 do not divide p then polynomials of degree 3 and 4 do not divide p? If polynomilas of degree 3 and 4 do not divide p then polynomials of degree 1 and 2 do not divide p? === Subject: Re: irreducible polynomial of degree 5. > Let p be a polynomial of degree 5 with coefficients in Q. > If polynomials of degree 1 and 2 do not divide p then polynomials of degree > 3 and 4 do not divide p? > If polynomilas of degree 3 and 4 do not divide p then polynomials of degree > 1 and 2 do not divide p? > Yes to both. The contraposits are easier. === Subject: probabilistic model for p_i (mod 3) and entropy In a cross-posted thread ``JSH: Randomness debate, some ground rules, > non_overlapping_prime_mod_k_tuple(k=2, p=3, limit=10**9) (1, 1) 5657771 (1, 2) 7051254 (2, 1) 7055917 (2, 2) 5658824 chisq 306302.383232 with 3 degrees of freedom The probabilistic model I have in mind is one where for i>=1, X_i takes values in {1,2}, with Prob(X_1 = 1} = K, 0reversal, S->sustained]. One sees that, whether X_i is 1 or 2, the probability of a reversal is always 0.555, and the probability of a sustained is always 0.445. Also, it seems to me that X_1 and the derived r.v.'s of type sustained or reversal are all independent. So I use an addition formula for the entropy of {X_1, ... X_N} in the form: H(X_1) + H(Y_2) + ... H(Y_N), where Y_{i+1} = R iff X_{i+1} =/= X_i, and Y_{i+1} = S otherwise. If this is valid, then the total entropy is a sum of entropies of Bernoulli trials. If p is the parameter for a Bernoulli trial Z, then H(Z) = p*log_2 (1/p) + (1-p)*log_2(1/(1-p), except for p=0 or 1, in which case H(Z) =0. So the entropy of {X_1, ... X_N} is then K*log_2(1/K) + (1-K)*log_2(1/(1-K) + + (N-1)* [0.555 log_2(1/0.555) + 0.445 log_2(1/0.445) ] or approximately K*log_2(1/K) + (1-K)*log_2(1/(1-K) + + (N-1)* (991/1000). As N grows larger, the above divided by N approaches 991/1000 or 0.991 (bits/symbol). Feedback appreciated. Bernier === Subject: Re: looking for a couple of formulas >I was loking for the teorem of Airy, of linear wave theory, mainly how >it was deducted, and was the consideration need to make the equations >linear I am not sure of the theorem you are asking about, but I have a document that describes the computation behind the formula for the Airy disk. See If that is not what you are looking for, please state more precisely what theorem you want. Rob Johnson take out the trash before replying === Subject: Re: looking for a couple of formulas it was deducted, and was the consideration need to make the equations > linear > alen Isenhart u#440483 I would try this question on sci.physics, or maybe one of the more advanced groups sci.math.research sci.physics.research and spell out the specific formulas that you have in mind. LH === Subject: Re: Who is Jesus Christ >> Good one, Bob ;-) >> Assuming this is an original: >> http://users.telenet.be/vdmoortel/dirk/Physics/ImmortalGems.html It seemed pretty obvious. I am sure others have thought of it, too. Sure, but not with *those* words ;-) Dirk Vdm === Subject: Re: Who is Jesus Christ > Hi to All, > My name is Islam, I am from Saudia . > I've seen many places of the world on TV screen and few that I've > visited for fun or/and business > As you know when we travel we meet a lot of different cultures and > people. > I found in many places' I've been to; that people stereotyped Islam > (that's my religion) they prejudge Muslims from they see in media > incidents. > Allow me to share with you here some information about Jesus Christ. > Muslims believe in Jesus Christ (peace and blessings of (Allah=God) be > upon Him as one of the mightiest messengers from (Allah=God). > We believe he is born miraculously by the word of Allah without father > And Marry (mar yam) is his mother the pious, pure female may Allah be > pleased with her. I also want to ask u a simple question. Who is Gabriel? Is he really an angel from God as he said to Mohammed ( that if Gabriel exists of course ), or he is some bad intellegent creature that fooled poor simple illetrate Mohammed who simplly believeg him . Gabriel is a suspecious creature, how do you want me to base my knoweldge about god on such an incognito creature, that might have bad intentions, Gabriel being so intellegent that he can speak in the miraculous way present in the qurran, doesn't mean that he necessarily came from God as you believe, it simply mean that Gabriel is an intellegent creature. it doesn't tell us if he was a good creature of a bad one. Intellegence is different from ethics. I believe that Islam is as christianity a couple of mythical material, that fool people helplesselly follow. Time should come were these people should open their eyes on the truth of these fallacies that raked their life apart. Zuhair === Subject: Re: Who is Jesus Christ > Hi to All, > My name is Islam, I am from Saudia . > I've seen many places of the world on TV screen and few that I've > visited for fun or/and business > As you know when we travel we meet a lot of different cultures and > people. > I found in many places' I've been to; that people stereotyped Islam > (that's my religion) they prejudge Muslims from they see in media > incidents. > Allow me to share with you here some information about Jesus Christ. > Muslims believe in Jesus Christ (peace and blessings of (Allah=God) be > upon Him as one of the mightiest messengers from (Allah=God). > We believe he is born miraculously by the word of Allah without father > And Marry (mar yam) is his mother the pious, pure female may Allah be > pleased with her. I also want to ask u a simple question. Who is Gabriel? What a coincidence! Just last night I was perusing Max Cannon's Red Meat comic strip when one of the archived titles caught my eye: Gabriels' Spit Trap. Spooky, eh? > Is he really an > angel from God as he said to Mohammed ( that if Gabriel exists of > course ), or he is some bad intellegent creature that fooled poor > simple illetrate Mohammed who simplly believeg him . Gabriel is a suspecious creature, how do you want me to base my > knoweldge about god on such an incognito creature, that might have bad > intentions, Gabriel being so intellegent that he can speak in the > miraculous way present in the qurran, doesn't mean that he necessarily > came from God as you believe, it simply mean that Gabriel is an > intellegent creature. it doesn't tell us if he was a good creature of a > bad one. Intellegence is different from ethics. I believe that Islam is as christianity a couple of mythical material, > that fool people helplesselly follow. Time should come were these > people should open their eyes on the truth of these fallacies that > raked their life apart. Zuhair === Subject: Re: Who is Jesus Christ I believe that Islam is as christianity a couple of mythical material, > that fool people helplesselly follow. Time should come were these > people should open their eyes on the truth of these fallacies that > raked their life apart. Religion is mostly nonsense. Some of it contains a nucleus of workable ethics. For example, Judaism changed sufficiently to permit Rabinnical Opinion (as in the Talmud) to jetison most of the theological nonsnese and leave a residue of working ethics. And that is why Juduaism has survived as long as it has even in a climate of hostility. The next time you are in Rome and see the Arch (Judea Capta) ask yourself where is the Imperium Romanum. There is nothing wrong with myth put to useful labor. Bob Kolker === Subject: Re: Who is Jesus Christ <4s7hkrFucjbqU1@mid.individual.net > I believe that Islam is as christianity a couple of mythical material, > that fool people helplesselly follow. Time should come were these > people should open their eyes on the truth of these fallacies that > raked their life apart. Religion is mostly nonsense. Some of it contains a nucleus of workable > ethics. For example, Judaism changed sufficiently to permit Rabinnical > Opinion (as in the Talmud) to jetison most of the theological nonsnese > and leave a residue of working ethics. And that is why Juduaism has > survived as long as it has even in a climate of hostility. The next time > you are in Rome and see the Arch (Judea Capta) ask yourself where > is the Imperium Romanum. There is nothing wrong with myth put to useful labor. Bob Kolker I agree, but you see Bob, the problem with myths is that they have intrinsic way of evolution, that is most of the time non practical. Even if the original founder of the myth , had good intentions, and his sole intention was ethical, yet the ideation itself has its own way of development that is very difficult to contain , especially when a myth is believed as a religion. it usually turns to be disasterous at the end. Zuhair === Subject: Re: Who is Jesus Christ > I agree, but you see Bob, the problem with myths is that they have > intrinsic way of evolution, that is most of the time non practical. > Even if the original founder of the myth , had good intentions, and his > sole intention was ethical, yet the ideation itself has its own way of > development that is very difficult to contain , especially when a myth > is believed as a religion. it usually turns to be disasterous at the > end. It worked fairly well for Judaism. Jews have survived over 3000 years and against steep odds. It remains to be seen what will become of Christianity and Islam. Bob Kolker === Subject: Re: Who is Jesus Christ <4s7hkrFucjbqU1@mid.individual.net> <4s7k5jFub0c1U2@mid.individual.net I agree, but you see Bob, the problem with myths is that they have > intrinsic way of evolution, that is most of the time non practical. > Even if the original founder of the myth , had good intentions, and his > sole intention was ethical, yet the ideation itself has its own way of > development that is very difficult to contain , especially when a myth > is believed as a religion. it usually turns to be disasterous at the > end. It worked fairly well for Judaism. Jews have survived over 3000 years > and against steep odds. It remains to be seen what will become of > Christianity and Islam. Bob Kolker Well, there is really no great difference between these three religions, All are myths. I think that people should base their ethical standards on objective material. for example the idea of mutual benefit. IF a metaphysical being is necessary for some to believe in, then I think , it should be contemplated only in general manner, that this being is God, the final aim of that life is towards the good, were good is a sort of cosmic mutual benefit. etc , etc,......... Respecting others and viewing your interests within the promotion those of others, and the others viewin their interest towards promoting your interest, i.e Mutual benefit. I think this is the nucleus of every good moral system. and perhaps even of every good metaphysical speculation necessary for people to belief ( i.e. Religion ). IF a good metaphysical being should exist, then he should approve that. And he should respect us when we say that he is otherwise incognito. since he chose be lie behind curtains, for purpose best known for himself, then we have no position to speak of him other than that of him being generally good and support mutual benefit, Love , Happyess, kindness, etc......... Zuhair === Subject: Re: Who is Jesus Christ <4s6736Fu3spoU2@mid.individual.net> <4s7425Frc9vuU1@mid.individual.net Good one, Bob ;-) > Assuming this is an original: > http://users.telenet.be/vdmoortel/dirk/Physics/ImmortalGems.html It seemed pretty obvious. I am sure others have thought of it, too. Bob Kolker Jesus Christ is only a myth.Bob, there is no mairry nor any of the rubbish surrounding these stories. The truth is that if God exists, then we know nothing of his will, we only can hope that he is a Good God with some mysterious wisdom that we cannot understand, that somehow it will turn good for the good people and the evil ones shall perish. If he doesn't exist, we can still hope for something in this nature to become good one day, that justace may finally prevail, and we can make a fantasic hope that the memory of this ultimate good nature would never forget us, that we shall be in a good place after all these misseries we are living through.Though there is no scientific evidence of this optimisim, but I think if one decides to live this life and bring children into it, then he is up to formulate such optimimism. Otherwise better leave, or live alone in a pessemistic grave he calls it life. Zuhair === Subject: Re: Liouvilles formula <455d0b00$0$176$edfadb0f@dread11.news.tele.dkSorry please ignore the above which was erroneously considered for a >different situation. There are two variables and arc length s is computed by direct >integration using partial derivatives and the result must in general >involve u and v invariably because it is a 2-D, two parametric surface. >Arc length can be computed by classical comptations. ds^2 = E du^2 + 2 >F du dv + G dv^2. If u = const lines are considered then du = 0 ,s(v) = >integral sqrt(G) dv. G involves partial derivatives of surface >co-ordinates with respect to v. The following example is of a sphere >X = a ( cos(u) cos(v), cos(u) sin(v), sin(u) ) with u = const latitude >circles (non-geodesic parallels except the equator) and v = const >longitude geodesic meridians.We compute arc length of the parallels. >ds = sqrt(G) dv = a cos(u) dv, s = a v cos(u). For the full equator arc >length is 2 pi a and is reduced by factor cos(u) to zero at the poles. In the above example, check s1 = a v cos(u), s2 = a u , by direct computation. > So you agree with my teacher and Shaums is wrong since Shaums forgot to > differentiate s1 with respect to v. May be,but best is to cross tally cases when grid is not orthogonal by direct and tangent vector normed methods. === Subject: Re: Liouvilles formula <455d0b00$0$176$edfadb0f@dread11.news.tele.dkSorry please ignore the above which was erroneously considered for a >different situation. There are two variables and arc length s is computed by direct >integration using partial derivatives and the result must in general >involve u and v invariably because it is a 2-D, two parametric surface. >Arc length can be computed by classical comptations. ds^2 = E du^2 + 2 >F du dv + G dv^2. If u = const lines are considered then du = 0 ,s(v) = >integral sqrt(G) dv. G involves partial derivatives of surface >co-ordinates with respect to v. The following example is of a sphere >X = a ( cos(u) cos(v), cos(u) sin(v), sin(u) ) with u = const latitude >circles (non-geodesic parallels except the equator) and v = const >longitude geodesic meridians.We compute arc length of the parallels. >ds = sqrt(G) dv = a cos(u) dv, s = a v cos(u). For the full equator arc >length is 2 pi a and is reduced by factor cos(u) to zero at the poles. In the above example, check s1 = a v cos(u), s2 = a u , by direct computation. > So you agree with my teacher and Shaums is wrong since Shaums forgot to > differentiate s1 with respect to v. May be,but best is to tally cases when grid is not orthogonal by direct and tangent vector normed methods. === Subject: CB: James Harris, you are own3d Sorry to feed that notorious troll James Harris here, but since he decided to insult me in a recent post, I will answer. 1. James Harris, your talent as a mathematician can only be described as zero. You haven't produced any useful mathematics in your whole life. There has not been one single post of yours on sci.math that would have been helpful to anybody. Your greatest achievement, the so-called prime-counting function, was invented by a real mathematician, Adrien-Marie Legendre, about 200 years ago, and you haven't even been able to understand improvements made to that method 100 years ago. Compared to me, you know absolutely nothing. And there are people posting here who are as far ahead of me in mathematics as I am ahead of you. 2. James Harris, as a programmer I can only classify you as unemployable. After seeing code that you have written, you wouldn't even make it to a job interview where I work, and that is just based on your programming skills. Personal traits are also important in any attempt to find employment, and with what we know about you, no amount of programming skills could secure you a job. 3. James Harris, at a personal level I believe that you are below the lowest of pond scum. I can remember you boasting about how you tried to get one excellent mathematician fired for some imagined insult. Not only are you despicable, you are even proud of it. 4. Now to your so-called challenge: The prime counting code that I posted three years ago ran about 1000 times faster than the fastest code you ever produced. That is ONE THOUSAND TIMES FASTER. You are completely owned. You are so far behind, you are not even a joke. Now for some real mathematics: Given that the Lagarias/Odlyzko/Miller algorithm calculates pi (N) in time roughly proportional to O (N^(2/3)), can it be modified to calculate pi (N(i)) for k different values N(i) <= N, 1 <= i <= k simultaneously, substantially faster than in O (k * N^(2/3)) steps ? As an example, if you wanted to calculate a table of values pi (k * 10^16) for all 1 <= k <= 10,000, can that be done substantially faster than using the Lagarias/Odlyzko/Miller algorithm k times? === Subject: Re: CB: James Harris, you are own3d > Sorry to feed that notorious troll James Harris here, but since he > decided to insult me in a recent post, I will answer. 1. James Harris, your talent as a mathematician can only be described > as zero. You haven't produced any useful mathematics in your whole > life. There has not been one single post of yours on sci.math that > would have been helpful to anybody. Your greatest achievement, the > so-called prime-counting function, was invented by a real > mathematician, Adrien-Marie Legendre, about 200 years ago, and you > haven't even been able to understand improvements made to that method > 100 years ago. Compared to me, you know absolutely nothing. And there > are people posting here who are as far ahead of me in mathematics as I > am ahead of you. 2. James Harris, as a programmer I can only classify you as > unemployable. After seeing code that you have written, you wouldn't > even make it to a job interview where I work, and that is just based on > your programming skills. Personal traits are also important in any > attempt to find employment, and with what we know about you, no amount > of programming skills could secure you a job. 3. James Harris, at a personal level I believe that you are below the > lowest of pond scum. I can remember you boasting about how you tried to > get one excellent mathematician fired for some imagined insult. Not > only are you despicable, you are even proud of it. 4. Now to your so-called challenge: The prime counting code that I > posted three years ago ran about 1000 times faster than the fastest > code you ever produced. That is ONE THOUSAND TIMES FASTER. You are > completely owned. You are so far behind, you are not even a joke. > See what I mean? Yup, Christian Bau is still an obnoxious little . But with that said, he still managed a lot more than anyone else here with other people's research, coding his own version of Odlyzko, Lagarias et. al. which is screamingly fast. Beat my code easily and handily. Can't anyone else here? James Harris === Subject: Re: CB: James Harris, you are own3d > See what I mean? Yup, Christian Bau is still an obnoxious little . I guess the big difference between you and him is that he is an obnoxious little who knows what he's talking about. === Subject: Re: CB: James Harris, you are own3d : See what I mean? Yup, Christian Bau is still an obnoxious little . : But with that said, he still managed a lot more than anyone else here : with other people's research, coding his own version of Odlyzko, : Lagarias et. al. which is screamingly fast. : Beat my code easily and handily. : Can't anyone else here? I'm not interested. You've made claims over the years, most of which have been shot down by various parties here. Christian shot this one down and made you look like a chump. I don't need to do it myself. Justin === Subject: Re: CB: James Harris, you are own3d > Sorry to feed that notorious troll James Harris here, but since he > decided to insult me in a recent post, I will answer. 1. James Harris, your talent as a mathematician can only be described > as zero. You haven't produced any useful mathematics in your whole > life. There has not been one single post of yours on sci.math that > would have been helpful to anybody. Your greatest achievement, the > so-called prime-counting function, was invented by a real > mathematician, Adrien-Marie Legendre, about 200 years ago, and you > haven't even been able to understand improvements made to that method > 100 years ago. Compared to me, you know absolutely nothing. And there > are people posting here who are as far ahead of me in mathematics as I > am ahead of you. 2. James Harris, as a programmer I can only classify you as > unemployable. After seeing code that you have written, you wouldn't > even make it to a job interview where I work, and that is just based on > your programming skills. Personal traits are also important in any > attempt to find employment, and with what we know about you, no amount > of programming skills could secure you a job. 3. James Harris, at a personal level I believe that you are below the > lowest of pond scum. I can remember you boasting about how you tried to > get one excellent mathematician fired for some imagined insult. Not > only are you despicable, you are even proud of it. 4. Now to your so-called challenge: The prime counting code that I > posted three years ago ran about 1000 times faster than the fastest > code you ever produced. That is ONE THOUSAND TIMES FASTER. You are > completely owned. You are so far behind, you are not even a joke. > See what I mean? Yup, Christian Bau is still an obnoxious little . But with that said, he still managed a lot more than anyone else here > with other people's research, coding his own version of Odlyzko, > Lagarias et. al. which is screamingly fast. Beat my code easily and handily. Can't anyone else here? > James Harris What's that got to do with anything? I think my implementation of the Lagarias-Miller-Odlyzko algorithm will beat yours at 10^15, but that's neither here nor there. I'm not a programmer. I don't know the tricks for making code fast. The real achievement is to come up with the algorithm in the first place. Yes, I am impressed that you can write code that will evaluate the function that fast for the small values, using such a bad algorithm. But much better algorithms are known and any competent programmer could write much faster code than yours. Christian already has. Presumably other people can't be bothered. Why would we care? There's plenty of software out there that's already been written for this purpose. === Subject: Re: JSH: Where is Christian Bau? Kind of wondering where that nasty person went? Not sure, but rumor is that the NSA grabbed him when they found out he was > lying about your research. They probably executed him. Funny that you are saying that... Last thing that I did at my previous job was writing some highly optimised code for AES-128 to run on an ARM processor, and right now I'm working on adding OpenSSL for authentication to a commercial product, but that's just coincidence. Still alive and well. === Subject: Probability and infinity I am unable to figure out what is the right answer to the following riddle. You have one dollar. You throw a die. If it comes up 6 you lose a dollar, otherwise you win a dollar. If you have zero dollars the game ends. If you have more than zero dollars the game continues and you throw another die. Rinse and repeat as often is necessary. The question: What is the probability that you will reach zero dollars? I have two ways of looking at it: 1. There is only one way in which the game can end and that is with you going bankrupt, so the probability of you reaching zero is 1. 2. The probability is larger that you move away from zero than towards it at each interval. So if a lot of these games are played, most won't end (in fact 4/5 if my calculation is correct). I don't know how to know which of these answers is correct (or if in fact any of them is correct). Help appreciated. === Subject: Re: Probability and infinity > I am unable to figure out what is the right answer to the following > riddle. You have one dollar. You throw a die. If it comes up 6 you lose a > dollar, otherwise you win a dollar. If you have zero dollars the game > ends. If you have more than zero dollars the game continues and you > throw another die. Rinse and repeat as often is necessary. The question: What is the probability that you will reach zero > dollars? I have two ways of looking at it: 1. There is only one way in which the game can end and that is with > you going bankrupt, so the probability of you reaching zero is 1. 2. The probability is larger that you move away from zero than towards > it at each interval. So if a lot of these games are played, most won't > end (in fact 4/5 if my calculation is correct). I don't know how to know which of these answers is correct (or if in > fact any of them is correct). Help appreciated. 4/5 is correct. For a simple random walk with absorbing barrier at 0 and probability of increase p the probability of going bust starting from k is ((1-p)/p)^k. if p > 1/2. If p <= 1/2 then you always go bust eventually. The proof is a little long for me to really want to type it all up in a post. You can probably find it by Google search for simple random walk or maybe gambler's ruin. If you really can't then ask. -- mike. === Subject: Re: Probability and infinity > I am unable to figure out what is the right answer to the following > riddle. You have one dollar. You throw a die. If it comes up 6 you lose a > dollar, otherwise you win a dollar. If you have zero dollars the game > ends. If you have more than zero dollars the game continues and you > throw another die. Rinse and repeat as often is necessary. The question: What is the probability that you will reach zero > dollars? I have two ways of looking at it: 1. There is only one way in which the game can end and that is with > you going bankrupt, so the probability of you reaching zero is 1. > That argument is not sufficient. The set of gameplays which go on for ever is nonempty, and it might have measure greater than zero. > 2. The probability is larger that you move away from zero than towards > it at each interval. So if a lot of these games are played, most won't > end (in fact 4/5 if my calculation is correct). > Yes, this argument is correct. I don't know how to calculate the probability exactly. It can be expressed as an infinite series but there might be a way to simplify it. > I don't know how to know which of these answers is correct (or if in > fact any of them is correct). Help appreciated. === Subject: Re: Big Bertha Thing blogs Big Bertha Thing lightcraft Cosmic Ray Series Possible Real World System Constructs http://web.onetel.com/~lance/liteship.html 14K Web page Astrophysics net ring Access site Newsgroup Reviews including uk.rec.gardening Lightcraft powered by lasers and microwaves v1.1 15 sep 99 greg goebel public domain Contents List:- 1.LASER LIGHTCRAFT 2.MICROWAVE LIGHTCRAFT 3.LASER LIGHTCRAFT IN TEST 4.COMMENTS, SOURCES, & REVISION HISTORY Big Bertha Thing property All pages are handcoded in HTML level 2. No pages are proprietary copyrighted HTML code copies. Such copies would be illegal. Such are in restraint of trade. Such are theft of intellectual property. Such close down the open web. All refugees from such are welcome. Such delete downloaded pages. Such download twice regardless. Such switch off the text only option. Such demand to be the default browser. Such yield control not even to power down. Lance peterpaul@bigberthathing.co.uk Big Bertha Thing Probe-B the Gravity Probe-B satellite launch. This is expected to prove the existence of a new force of nature, proposed by Einstein. Free Press Pack Download (11MB PDF) http://www.gravityprobeb.com/gpb_presskit.pdf There are one or two problems with this:- 1. Einstein never proposed this new force of nature. (Funding canard) 2. It is a magnetic force directly proportional to angular momentum. (Memomagnetism) 3. It is weak at Newtonian speeds and strong at near light speeds. 4. The big three in gyroscopes are Einstein, Professor Francis Everitt of Stanford University and Harold Crabtree M.A. 6. The relevant extract from the book is given below. (Spider published 4th March 1998) 7. They do not credit the costermonger, with the definitive experiment, which proves the existence of the new force of nature. 8. They do not accept that the novelty item was a valid experiment. 9. They believe that the spider wheel was magnetized. 10. The costermonger did not magnetize the spider wheel, he exploited a force of nature. 11. The costermonger discovered, applied and sold the first working application. 12. There is an eye-witness account from 1850 by the author, which kindled his life-long interest. 13. The description was published in 1909, in the same book. 14. The force of nature has been missing for 150 years, since its discovery. 15. The first modern publication on the subject is spider, on 4th March 1998. (Usenet) 16. Attribution and due credit are two of the failings of modern science. Big Bertha Thing spider Cosmic Ray Series Possible Real World System Constructs http://web.onetel.com/~lance/spider.html Access page JPG 11K Image Astrophysics net ring Access site Newsgroup Reviews including uk.rec.cycling Drawing of a clockwork spider wheel and hairpin. Extract from Introductory Chapter;- The Spider tops, which are frequently sold in the streets of London, consist of a heavy little disc mounted on a spindle (Fig. XIV.). When the disc has been set spinning a small curved piece of metal is placed to touch the toe, and at once begins to slide round it, first the side (a) in the figure, and then the side (b), the motion continuing backwards and forwards till the top comes to rest. The fact is that the toe is magnetic, and this being the case it is easy to see that the rolling of the toe on the side of the metal produces the motion. An Elementary Treatment of the Theory of Spinning Tops and Gyroscopic Motion. By Harold Crabtree M.A. Formerly Scholar of Pembroke College, Cambridge Assistant Master at Charterhouse Longmans, Green and Co. 1923 First Edition 1909 Second Edition 1914 New Impression 1923 (C) Copyright Lance 1998 Distribute complete and free of charge to comply. Big Bertha Thing fact Anything but a fact, changes the face of twentieth century science. 1. No iron moons and planetary cores. 2. No red shift measure of speed. 3. No Patrick Moore star at 95% the speed of light. 4. Muons arrive on earth. 5. Relativity is like an imaginary number; useful but not real. 6. Einstein-Haas gives a field strength 1/10000th the electric field. 8. Schroedinger is an approximation. Who has the wit to check the fact? === Subject: Re: I don't understand this! Sorry dont really understand what they're trying to say in wikipedia... So is 0.99999... really equal to 1? if that's true then i'll be able to write things like: 0.999... * 0.9999... = 1 sqrt (0.999...) = 1 1 + 1 = 1.9999.... etc... ? === Subject: Re: I don't understand this! hsxhua001 ha escrito: > Sorry dont really understand what they're trying to say in wikipedia... So is 0.99999... really equal to 1? if that's true then i'll be able to write things like: 0.999... * 0.9999... = 1 sqrt (0.999...) = 1 1 + 1 = 1.9999.... etc... ? ***************************************** Yes. You REALLY didn't understand what they're trying to say in the Tonio === Subject: Re: I don't understand this! Originator: pouya@localhost [hsxhua001 ] >Sorry dont really understand what they're trying to say in wikipedia... So is 0.99999... really equal to 1? if that's true then i'll be able to write things like: 0.999... * 0.9999... = 1 sqrt (0.999...) = 1 1 + 1 = 1.9999.... etc... ? > Yes. -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Defining Natural Numbers Hi all, Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the set of natural numbers without the use of any axioms ? pradyumna === Subject: Re: Defining Natural Numbers > Hi all, > Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the > set of natural numbers without the use of any axioms ? Why not keep it simpler? If you can't do it with D = {0, 1}, then a bigger set of symbols won't help. -- === Subject: Re: Defining Natural Numbers > Hi all, > Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the > set of natural numbers without the use of any axioms ? pradyumna I don't understand this question. === Subject: Re: Defining Natural Numbers > Hi all, > Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the > set of natural numbers without the use of any axioms ? pradyumna I don't understand this question. Let me put it this way: Let's say a person doesn't know anything about numbers. I give him ten lexical tokens 0, 1, 2, 3 etc. So all we have now is a set of symbols (not numbers) , S = {0, 1, 2, ...9 } Now we define a relationship amongst the elements of S so that an order such as 0 < 1, 5 < 3 comes into picture. (this may be explicit and comprehensive, which specifies every such binary relation among these ten symbols) Now, can we define another set of symbols S' = {0, 1, 2...9, 10, 11 ... 100, 101...} utilizing the set S (and some sort of concatenation) S' will have an infinite number of symbols. Once this is done, we might define a less than relationship among the elements of S' Taking this further, we might land up on the set of natural numbers, from where Q and R can be deduced following standard procedures. I'm not precise here. This may even be inadmissible, but I wish to develop an idea on these lines. I want to know if there is any resource that arrives at natural numbers this way (a book or a paper or a web-page) pradyumna === Subject: Re: Defining Natural Numbers > Hi all, > Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the > set of natural numbers without the use of any axioms ? pradyumna I don't understand this question. Let me put it this way: Let's say a person doesn't know anything about numbers. I give him ten > lexical tokens 0, 1, 2, 3 etc. So all we have now is a set of symbols (not numbers) , S = {0, 1, > 2, ...9 } Now we define a relationship amongst the elements of S so that an order > such as 0 < 1, 5 < 3 comes into picture. (this may be explicit > and comprehensive, which specifies every such binary relation among > these ten symbols) Now, can we define another set of symbols S' = {0, 1, 2...9, > 10, 11 ... 100, 101...} utilizing the set S (and some sort of > concatenation) S' will have an infinite number of symbols. Once this is done, we might define a less than relationship among the > elements of S' Taking this further, we might land up on the set of natural numbers, > from where Q and R can be deduced following standard procedures. I'm not precise here. This may even be inadmissible, but I wish to > develop an idea on these lines. I want to know if there is any resource > that arrives at natural numbers this way (a book or a paper or a > web-page) pradyumna It can be done strictly in terms of symbols, though we can't get very a good equivalent to real numbers without infinitely long symbols. I did it once, a long time ago, up through integers and rationals, but it was very ugly, and I did not keep my notes on it. === Subject: Re: Defining Natural Numbers > Hi all, > Given a set of symbols D = {0,1,2,3,4,5,6,7,8,9}, can we arrive at the > set of natural numbers without the use of any axioms ? pradyumna I don't understand this question. Let me put it this way: Let's say a person doesn't know anything about numbers. I give him ten > lexical tokens 0, 1, 2, 3 etc. So all we have now is a set of symbols (not numbers) , S = {0, 1, > 2, ...9 } Now we define a relationship amongst the elements of S so that an order > such as 0 < 1, 5 < 3 comes into picture. (this may be explicit > and comprehensive, which specifies every such binary relation among > these ten symbols) Now, can we define another set of symbols S' = {0, 1, 2...9, > 10, 11 ... 100, 101...} utilizing the set S (and some sort of > concatenation) S' will have an infinite number of symbols. Once this is done, we might define a less than relationship among the > elements of S' Taking this further, we might land up on the set of natural numbers, > from where Q and R can be deduced following standard procedures. I'm not precise here. This may even be inadmissible, but I wish to > develop an idea on these lines. I want to know if there is any resource > that arrives at natural numbers this way (a book or a paper or a > web-page) pradyumna I don't understand how you expect to do it without any axioms. You're basically asking whether the natural numbers can be defined in terms of the digits you've given. Well, provided you've got enough set theory to work with, you could probably do that. But you need to work in some theory, and it's got to have axioms. === Subject: Re: Defining Natural Numbers > I don't understand how you expect to do it without any axioms. You're > basically asking whether the natural numbers can be defined in terms of > the digits you've given. Well, provided you've got enough set theory to > work with, you could probably do that. But you need to work in some > theory, and it's got to have axioms. Well, provided we got enough set theory, we don't need *any* of those symbols. The only thing we need is an empty set, right? (Von Neumann Numerals, anyone). === Subject: Re: Simplicity of PSL(2,q) > What is the easiest way of proving this (when it is true), > please. > Can one use Lie Algebra theory? > I think I would recommend studying the proof of the more general result that PSL(n,K) is simple for all n > 1 and all fields K, except for n=2, |K|=2,3. It goes roughly as follows. 1. Prove SL(n,K) is generated by transvections. 2. Prove that all transvections are commutators (except for n=2, |K|=2,3), and hence SL(n,K) is perfect. 3. The final step is a short but tricky argument using the fact that PSL(n,K) acts 2-transitively on the 1-dimensional subspaces of the natural module for SL(n,K). I do not know of any argument for PSL(2,q) which is substantially quicker than this, unless you want to use the classification by Burnsaide of all subgroups of PSL(2,q). Derek Holt. === Subject: Re: Simplicity of PSL(2,q) >> What is the easiest way of proving this (when it is true), >> please. >> Can one use Lie Algebra theory? I think I would recommend studying the proof of the more general result > that PSL(n,K) is simple for all n > 1 and all fields K, except for n=2, > |K|=2,3. It goes roughly as follows. 1. Prove SL(n,K) is generated by transvections. 2. Prove that all transvections are commutators (except for n=2, > |K|=2,3), and hence SL(n,K) is perfect. 3. The final step is a short but tricky argument using the fact that > PSL(n,K) acts 2-transitively on the 1-dimensional subspaces of the > natural module for SL(n,K). I shall study that. I think the last part is the bit I found complicated, but it wasn't expressed in your way. > I do not know of any argument for PSL(2,q) which is substantially > quicker than this, unless you want to use the classification by > Burnsaide of all subgroups of PSL(2,q). I didn't know Burnside classified these subgroups. It struck me that it shouldn't be too difficult to determine the classes, and prove the result that way. Also, many years ago I read vol 3 of Chevalley's series on Lie groups, on Algebraic Groups, and my recollection is that an algebraic group over any field is simple if its Lie algebra is simple. But I'm sure it cannot be that simple (no pun intended). I also wondered if one could not use some version of Coxeter's approach, with the group defined by reflections, or at least involutions. But I did realize from my limited reading on the subject that transvections seem the way to go. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Simplicity of PSL(2,q) Can you please help me out, since I am not sure what you mean by PSL(2,q): The group PSL(2) over a field with q elements (q a prime power) or over Z/qZ or even something else? === Subject: Re: Simplicity of PSL(2,q) > Can you please help me out, since I am not sure what you mean by > PSL(2,q): The group PSL(2) over a field with q elements (q a prime > power) or over Z/qZ or even something else? I mean the group PSL(2,F_q) over the field F_q with q elements (where q must be a prime power, as you say). More precisely, SL(n,q) is the group of nxn matrices over F_q with determinant 1, and PSL(n,q) is the quotient of SL(n,q) by its centre, consisting of the multiples tI of the identity with t^n = 1. I think the notation is reasonably standard. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: An unusual measurable set I am trying to find a (Lebesgue) measurable set E such that the function f(r)= m(E cap I)/m(I) satisfies liminf_{r>0} f(r)=0 and limsup_{r>0} f(r)=1. Here I=(-r,r) and m is the Lebesgue measure. === Subject: Re: An unusual measurable set > I am trying to find a (Lebesgue) measurable set E such that the function > f(r)= m(E cap I)/m(I) no r on the RHS? > satisfies > liminf_{r>0} f(r)=0 and limsup_{r>0} f(r)=1. Here I=(-r,r) and m is the > Lebesgue > measure. Show there is a sequence b_1 > a_1 > b_2 > a_2 > ... -> 0, such that (b_n - a_n)/b_n -> 1 and a_n/b_(n+1) -> 0. Set E = U {a_n < |x| < b_n}, and consider f(b_n) and f(a_n). === Subject: Re: An unusual measurable set > I am trying to find a (Lebesgue) measurable set E such that the function > f(r)= m(E cap I)/m(I) satisfies > liminf_{r>0} f(r)=0 and limsup_{r>0} f(r)=1. Here I=(-r,r) and m is the > Lebesgue > measure. How about this: Let E = {x : 1 / n! > |x| > 1 / (n + 1)! for some even n}. Then, for even n, f(1 / n!) > n / (n + 1), and for n odd, f(1 / n!) < 1 / (n + 1). === Subject: Re: An unusual measurable set > I am trying to find a (Lebesgue) measurable set E such that the function > f(r)= m(E cap I)/m(I) satisfies > liminf_{r>0} f(r)=0 and limsup_{r>0} f(r)=1. Here I=(-r,r) and m is the > Lebesgue > measure. Construct it as a countable union of intervals, clustering at 0. === Subject: JSH: So what's your school? Given all the hostile arguing here it is worth getting information about educations. So here is mine and I hope posters who argue with me will give information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that I was part of Duke University's Talent Identification Program (TIP) as a teenager, where I took geometry and briefly abstract algebra in their summer program one year, and structured C, taught by an IBM researcher from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant backgrounds. James Harris === Subject: Re: JSH: So what's your school? > Given all the hostile arguing here it is worth getting information > about educations. So here is mine and I hope posters who argue with me will give > information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that > I was part of Duke University's Talent Identification Program (TIP) as > a teenager, where I took geometry and briefly abstract algebra in their > summer program one year, and structured C, taught by an IBM researcher > from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant > backgrounds. > James Harris I have an Associates's Degree in Electronic Engineering Technology from DeVry. The math classes I took (calculus, differential equations) are long forgotten. Except for Boolean Algebra. I specialized in digital electronics and became the world's greatest tester/trouble-shooter. First with hardware and then with software. Go over to comp.lang.python and search on gmpy+mensanator and see how I solved the divm() memory leak simply by perusing the source code. I had no way to compile and verify the problem, but from the source code and careful observation of symptoms, my guess as to what was wrong turned out to be spot on and now that library has been repaired. That is a better accomplishment than your pathetic Class Viewer. Currently, I do database management for a geotechnical consulting firm and work on environmental remediation projects. That means cleaning up the legacy of pollution left behind after a century of operations by oil refineries and steel mills. I, at least, am making a positive contribution to society. What have you done? And one last item. _My_ math paper Blueprint for Failure: How to Construct a Counterexample to the Collatz Conjecture hasn't been yanked from S.A.T.O. Volume 5.3. (2006) http://home.zonnet.nl/galien8 === Subject: Re: JSH: So what's your school? : Other posters would please give their education and math relevant : backgrounds. I have a PhD. in pure mathematics (geometry) and twenty-five years of programming experience starting at age 11. Justin === Subject: Re: JSH: So what's your school? > So here is mine and I hope posters who argue with me will give > information about their educations. I finished high-school, I think. But then as I don't argue with you it's probably irrelevant. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: JSH: So what's your school? > Given all the hostile arguing here it is worth getting information > about educations. So here is mine and I hope posters who argue with me will give > information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that > I was part of Duke University's Talent Identification Program (TIP) as > a teenager, where I took geometry and briefly abstract algebra in their > summer program one year, and structured C, taught by an IBM researcher > from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant > backgrounds. > James Harris You were 'gifted' 20 years ago (or more). You developed Narcisstic Personality Disorder, for which you've not been willing to get treatment. Since the onset of the disease (at least 11 years ago), you've done nothing, acheived nothing and wasted your life. You can still get help. Or you can just keep living this way. Not a lot we can do about it. === Subject: Re: JSH: So what's your school? > I have been diagnosed as gifted much of my life. It is just a word. > James Harris You've been diagnozed as manic-depressive personality with some paranoid tendencies, at least for the last 10 year. It is *not* just a word. === Subject: Re: So what's your school? > Given all the hostile arguing here it is worth getting information > about educations. So here is mine and I hope posters who argue with me will give > information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that > I was part of Duke University's Talent Identification Program (TIP) as > a teenager, where I took geometry and briefly abstract algebra in their > summer program one year, and structured C, taught by an IBM researcher > from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant > backgrounds. > James Harris > I have an Associate's Degree in Mathematics and am finishing my Bachelor's in Meteorology and Mathematics. My main Mathematical Interests are Differential Equations, Convexity Theory, and Regression. I also have programming back ground in C. Here's a laundry list of classes I've taken: Calculus I-IV Discrete Math Ordinary Differential Equations Partial Differential Equations Linear Algebra Statistical Methods Complex Variables Convexity Theory Probability Theory (taking currently) Dave === Subject: Re: JSH: So what's your school? > Given all the hostile arguing here it is worth getting information > about educations. So here is mine and I hope posters who argue with me will give > information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that > I was part of Duke University's Talent Identification Program (TIP) as > a teenager, where I took geometry and briefly abstract algebra in their > summer program one year, and structured C, taught by an IBM researcher > from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant > backgrounds. > James Harris I represented Australia in the International Maths Olympiad when I was in high-school, in 1992 and 1993. I achieved a Bronze Medal on both occasions. I have a first-class Honours degree in Pure Mathematics from the University of New South Wales. I am currently working on a Ph.D. === Subject: Re: JSH: So what's your school? > Given all the hostile arguing here it is worth getting information >> about educations. >> So here is mine and I hope posters who argue with me will give >> information about their educations. >> I have a B.Sc. in physics from Vanderbilt University. Previous to that >> I was part of Duke University's Talent Identification Program (TIP) as >> a teenager, where I took geometry and briefly abstract algebra in their >> summer program one year, and structured C, taught by an IBM researcher >> from the IBM Research Triangle, the next. >> I have been diagnosed as gifted much of my life. It is just a word. >> Other posters would please give their education and math relevant >> backgrounds. >> James Harris I represented Australia in the International Maths Olympiad when I was > in high-school, in 1992 and 1993. I achieved a Bronze Medal on both > occasions. I have a first-class Honours degree in Pure Mathematics from the > University of New South Wales. I am currently working on a Ph.D. I hope you realize that, unless you behave, he's going to write a letter to your University and explain to them why you shouldn't get that Pd.D ? Dirk Vdm === Subject: Re: JSH: So what's your school? > Given all the hostile arguing here it is worth getting information >> about educations. >> So here is mine and I hope posters who argue with me will give >> information about their educations. >> I have a B.Sc. in physics from Vanderbilt University. Previous to that >> I was part of Duke University's Talent Identification Program (TIP) as >> a teenager, where I took geometry and briefly abstract algebra in their >> summer program one year, and structured C, taught by an IBM researcher >> from the IBM Research Triangle, the next. >> I have been diagnosed as gifted much of my life. It is just a word. >> Other posters would please give their education and math relevant >> backgrounds. >> James Harris I represented Australia in the International Maths Olympiad when I was > in high-school, in 1992 and 1993. I achieved a Bronze Medal on both > occasions. I have a first-class Honours degree in Pure Mathematics from the > University of New South Wales. I am currently working on a Ph.D. I hope you realize that, unless you behave, he's going to > write a letter to your University and explain to them why > you shouldn't get that Pd.D ? > Could be a problem! > Dirk Vdm === Subject: Re: JSH: So what's your school? Given all the hostile arguing here it is worth getting information > about educations. So here is mine and I hope posters who argue with me will give > information about their educations. I have a B.Sc. in physics from Vanderbilt University. Previous to that > I was part of Duke University's Talent Identification Program (TIP) as > a teenager, where I took geometry and briefly abstract algebra in their > summer program one year, and structured C, taught by an IBM researcher > from the IBM Research Triangle, the next. I have been diagnosed as gifted much of my life. It is just a word. Other posters would please give their education and math relevant > backgrounds. > James Harris >> I represented Australia in the International Maths Olympiad when I was >> in high-school, in 1992 and 1993. I achieved a Bronze Medal on both >> occasions. >> I have a first-class Honours degree in Pure Mathematics from the >> University of New South Wales. I am currently working on a Ph.D. >> I hope you realize that, unless you behave, he's going to >> write a letter to your University and explain to them why >> you shouldn't get that Ph.D ? > > Could be a problem! Unless you behave, of course ;-) Dirk Vdm === Subject: Re: Concise prime counting functions, where? research is the assertion that it is well-known. However, I JUST did a survey of math websites doing a search on prime > counting and saw nothing like the following concise prime counting > function, which is mine. My prime counting function in its sieve form is as follows. With natural numbers x and n, where p_i is the i_th prime: P(x,n) = x - 1 - sum for i=1 to n of {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt(x) then n is reset to that count. That is very concise, and it is in one line where you have the > summation with my prime counting function in sieve form recursively > calling itself. It has been noted that it can be directly related to the phi function > which usually is discussed with Legendre's Method. But, um, I can see nothing out there on the web that gives anything > like my prime counting function, so, what if? What if there isn't anything on the web because no one figured it out > before? > [...] There is a lot of mathematics on the Web (even after discounting > multi-megabytes of pi). There is also a lot of math > in books and journals. Algorithms for computing > pi(n) for large values of n would fall under computational > number theory. A nonrecursive formula using only the four arithmetic operations and > the floor function for pi(n) and the nth prime (by Jonathan Sondow and > Sebastian Martin Ruiz) can be found in http://xxx.lanl.gov/abs/math.NT/0210312 > I might check it out, but you're dodging the actual question. My prime counting function: With natural numbers x and n, where p_i is the i_th prime: P(x,n) = x - 1 - sum for i=1 to n of {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count. The claim put forward is that mathematicians already had it, so I'm asking for citations of something that short. Otherwise, if math people had it already, they have deliberately NOT used a concise expression, but that really tests the imagination and credibility of rational people who would think that somewhere in the voluminous record keeping of the math world someone would have expressed the prime counting function that way--if it were actually known before me. > Of course, JSH won't be interested in it. And if someone _did_ cite something, would JSH be able to find it? > Would he actually take the time to look for it? I think not. > I am curious as it's MY research and claims of it not being unique to me are for that reason of interest to me. > Back in March, I posted the following: I have in front of me two papers: J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing $pi(x)$: > The Meissel-Lehmer method, Math. Comp. 44 (1985), 537--560. M. DELEGLISE AND J. RIVAT, COMPUTING $pi(x)$: THE MEISSEL, LEHMER, > LAGARIAS, MILLER, ODLYZKO METHOD, MATHEMATICS OF COMPUTATION, > Volume 65, Number 213 (January 1996), Pages 235--245. Your formulas (in an in inefficient form) appear on p. 540 of the > Lagarias paper and p. 236 of the Deleglise and Rivat paper. > And I checked and you're wrong. Why don't you just post what is actually there? Then readers can compare to: With natural numbers x and n, where p_i is the i_th prime: P(x,n) = x - 1 - sum for i=1 to n of {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count. > Of course, I'm on his black list, so he won't be reading this post. --- You got that wrong, and how much else did you get wrong too? I HATE how you posters manage to dodge ever actually delivering on your claims and the newsgroups either don't care--as in, nobody is reading--or they just don't care about the truth. I just hope nobody is reading as it amazes me that people can come here and lie about such direct things, never cite anything that supports them, and the Usenet math community just ignore a concise mathematical expression for counting primes. Usenet is supposed to be the place where you can go when the mainstream blocks you for political reasons. But with math newsgroups, Usenet is just wannabe math mainstream, and can lie with everyone else, ignore the most basic things, and then people go on as if they represent real intelligence. James Harris === Subject: Convolution ??? Hi - convolution of series and I thought I had it right, that this meant (for the case of two series): given 2 series A = a0 + a1 + a2 + ... B = b0 + b1 + b2 + ... it were C = a0*b0 + a1*b1 + ... But now, looking at mathworld and wikipedia I seem to have missed the point. Is there another name for the above operation aside vectorial-crossproduct (which I would have attributed to this)? Or am I missing something. Gottfried Helms === Subject: Re: Convolution ??? > Hi - convolution of series and I thought I had it right, that this > meant (for the case of two series): given 2 series A = a0 + a1 + a2 + ... > B = b0 + b1 + b2 + ... it were > C = a0*b0 + a1*b1 + ... But now, looking at mathworld and wikipedia I seem to > have missed the point. > Is there another name for the above operation aside > vectorial-crossproduct (which I would have attributed > to this)? > Or am I missing something. Gottfried Helms Wrong again, honey. C (as defined above) a dot product of A and B, and I guess we have to assume that both ||A|| and ||B|| are finite. === Subject: Re: Convolution ??? >Hi - convolution of series and I thought I had it right, that this > meant (for the case of two series): given 2 series A = a0 + a1 + a2 + ... > B = b0 + b1 + b2 + ... it were > C = a0*b0 + a1*b1 + ... But now, looking at mathworld and wikipedia I seem to > have missed the point. > Is there another name for the above operation aside > vectorial-crossproduct (which I would have attributed > to this)? > Or am I missing something. Your definition of C would usually be called the dot or inner product of the sequences (a0,a1,...) and (b0,b1,...). The convolution of these two sequences, on the other hand, would be another sequence rather than a number, with the k-th term given by: a0 bk + a1 b_{k-1} + ... + ak b0 Convolutions of sequences usually come up in the context of Fourier series on the unit circle. In this case, the sequences extend infinitely in both directions (i.e. (...,a_{-2},a_{-1},a0,a1,a2,...) and similarly for the b's) and each term in the convolution of the sequences is given by an infinite sum. -- Daniel Mayost === Subject: Re: Convolution ??? Am 18.11.2006 18:28 schrieb Gottfried Helms: > Hi - convolution of series and I thought I had it right, that this > meant (for the case of two series): given 2 series A = a0 + a1 + a2 + ... > B = b0 + b1 + b2 + ... it were > C = a0*b0 + a1*b1 + ... But now, looking at mathworld and wikipedia I seem to > have missed the point. > Is there another name for the above operation aside > vectorial-crossproduct (which I would have attributed > to this)? > Or am I missing something. Gottfried Helms Well, after more intense reading of the online-encyclopedias I now understand it differs at least by inverting of the indices (and extending to negative indices as well). But I think, there was also a terminus technicus (differnt from vectorial crossproduct) used for the above operation. What had it just been... Gottfried Helms === Subject: Chi Square Goes to Normal Distribution After doing a bit of research on the internet, I have found out that the Chi Square distribution goes to the Normal Distribution as n degrees of freedom go to infinity. How can this be shown? What is the significance of this observation? === Subject: Re: Chi Square Goes to Normal Distribution >After doing a bit of research on the internet, I have found out that >the Chi Square distribution goes to the Normal Distribution as n >degrees of freedom go to infinity. How can this be shown? What is the >significance of this observation? > The chi-squared distribution with n degrees of freedom is the distribution of the sum of the squares of n independent normal random variables. TheCentral Limit Theorem tells us that the distribution of the sum of i.i.d. random variables approaches normal as the number of summands grows large. The significance? Use this approximation where appropriate. E.g., for large sample size, the the sample variance from a normal population has roughly normal distribution. I think this asymptotic holds for any finite-variance population. Also, in tests which call for the chi-squared distribution, you may use the normal table (if you work with tables) with large degrees of freedom. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Chi Square Goes to Normal Distribution <455F4DF0.7050303@netscape.net> Stephen... May I offer a small, but useful, suggestion. With approx 5 df or more the median of the Chi-sq distribution is close to the number of df in the data. (I know this is a bit crude, but it's close enough for now). So... if your calculated value of Chi-sq is less than the df from which those data spawned, then your observed value of Chi-sq is nowhere close to significant. Even if your calculated value of Chi-sq is just little greater than the number of df, your calculated (or observed) value of Chi-sq is still far from being significant. In other words, unless your value of Chi-sq is notably larger than the number of df from which it spawned... then not significant. If you can locate a table of the percentiles of Chi-sq/df, then locate the P50 point (50% point) and you'll find: df = 1 P50 = 0.455 df = 2 P50 = 0.693 df = 3 P50 = 0.789 df = 4 P50 = 0.839 df = 5 P50 = 0.870 df = 8 P50 = 0.918 df = 12 P50 = 0.945 df = 60 P50 = 0.989 df = 500 P50 = 0.999 and converges to 1.000 So if you are in a meeting or traveling and don't have a Chi-sq table nearby, this insight alone is often sufficient for the moment. Of course if the observed (calculated) value of Chi-sq is substantially larger than the df, then you will probably need a Chi-sq table to render the final judgment. This is one of those nice to know things... valuable when you need it. After doing a bit of research on the internet, I have found out that >the Chi Square distribution goes to the Normal Distribution as n >degrees of freedom go to infinity. How can this be shown? What is the >significance of this observation? > The chi-squared distribution with n degrees of freedom is the > distribution of the sum of the squares of n independent normal random > variables. TheCentral Limit Theorem tells us that the distribution of > the sum of i.i.d. random variables approaches normal as the number of > summands grows large. The significance? Use this approximation where appropriate. E.g., for > large sample size, the the sample variance from a normal population has > roughly normal distribution. I think this asymptotic holds for any > finite-variance population. Also, in tests which call for the > chi-squared distribution, you may use the normal table (if you work with > tables) with large degrees of freedom. -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Chi Square Goes to Normal Distribution >After doing a bit of research on the internet, I have found out that >the Chi Square distribution goes to the Normal Distribution as n >degrees of freedom go to infinity. How can this be shown? What is the >significance of this observation? > To show that the distribution approaches normal, use the central limit theorem along with the fact that with n degrees of freedom, the chi-square random variable is the sum of n iid random variables. -- Daniel Mayost === Subject: Re: Ode on A Grecian Urn? <4s770rFtut60U1@mid.individual.net >Okay, what's the relationship between Pythargoras' Theorem and Keats' >>Ode on A Grecian Urn?? > They both describe a high pot n' youth. God will punish you for that. > Sorry, my keyboard has a cold. And, hey, God loves puns. -- Neo D'Artagnan === Subject: Re: Ode on A Grecian Urn? <4s770rFtut60U1@mid.individual.net >Okay, what's the relationship between Pythargoras' Theorem and Keats' >>Ode on A Grecian Urn?? > They both describe a high pot n' youth. God will punish you for that. Sorry, my keyboard has a cold. And, hey, God loves puns. But only those that He makes. God takes a dim view of those who think they His peers. -- > Neo D'Artagnan === Subject: New Lottery in Michigan Tickets cost $20, and only 500,000 are printed, with a payoff of $1 million. A quick calculation shows an expectation value of $2 for the payoff, leaving you with a net loss of $18. In other words, they keep 90% of the money and pay back 10%. Those are about the worst odds I can imagine. Somewhere I heard that the odds at casinos are stacked in favor of the house to the tune of about 6% (not including card counting), but suddenly I'm skeptical of that number. Does anybody have any hard numbers that I can plug into the simple expectation value formula to allow me to calculate the odds of, say, the slot machines, which look like they should be simple enough to analyze quickly, easily and completely? === Subject: Re: New Lottery in Michigan Tickets cost $20, and only 500,000 are printed, with a payoff of $1 > million. A quick calculation shows an expectation value of $2 for the > payoff, leaving you with a net loss of $18. In other words, they keep 90% > of the money and pay back 10%. Those are about the worst odds I can > imagine. Somewhere I heard that the odds at casinos are stacked in favor of the > house to the tune of about 6% (not including card counting), but suddenly > I'm skeptical of that number. Does anybody have any hard numbers that I > can plug into the simple expectation value formula to allow me to > calculate > the odds of, say, the slot machines, which look like they should be simple > enough to analyze quickly, easily and completely? > Seems to me that the Vegas casinos brag about their rates, claiming, e.g. 98% payout etc. Though they may twist the math. And, the actual percentage will vary based on what game you play and somewhat with house rules. If you look up lotto 6/49 (wclc.com or similar) I believe they give the entire breakdown. Other lottos might as well. a 10% payback sounds horrible. I know someone who works at a company that produces a fair chunk of the world's scratch tickets. I think the payout rate was typically along the lines of 60%. There are also a lot of 'free play' prizes that go out, which you may want to figure into your calcs (you can buy a ticket that wins a free play then win money off the free play). === Subject: Re: Moebius Band is not homeomorphic with a Torus > I don't use Mathematica, but tell me if the following is correct. Define > a function f:R^2 -> R^3 by > f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), > ( cos(t/2 + k Pi/m) + 3 ) sin(t), > sin(t/2 + k Pi/m ) ). You are asking whether if we restrict f to {(t,k)| 0 < t < 4 Pi, 1 < k < 2m}, it is a homeomorphism. Is that your question? No, my question is not about the character of functional dependence > f(t, k) but about the flexibility of the two parameter domain, the > rubber sheet area parameter domain, if you will. The question again: For f (t, k) and for all arbitrary parameter domain sets (in your > notation) : {(t, k) | tmin < t < tmax, kmin < k < kmax}, where t's are real and k's are integers, do we have a set of surfaces > with valid homeomorphism among them? I don't understand what you are asking. I seem to be asking if something > is a homeomorphism. A homeomorphism is a map between two surfaces. So, > what are your two surfaces and what is the map? And, what does f have to > do with it? The two surfaces are portions of the same surface f( k,m) = > (x,y,z).Mapping is possible between different m values. f(k,m)=(x,y,z) doesn't make much sense. What are k and m? Once again, the following is explained. *********************************************************************** > For 2D patches in 3D Space. A patch is mapped onto another patch in a homeomorphism. A patch of such a surface has 3 coordinate transformations for x,y, z > and 2 parameter domains for th and k. What are th and k? A set of tori considered here has > a full parametric representation as: x=(a Cos[th/2+k Pi/m]+b) Cos[th]; y=(a Cos[th/2+k Pi/m]+b) Sin[th]; z=a > Sin[th/2+k Pi/m]; > [{x,y,z},{th,0,4 Pi},{k,1,2 m}] or [f(th,k), {th,0,4 Pi},{k,1,2 m}] How do I get one torus from this set? Your questions are addressed earlier. However, shall once again > clarify. m -> Infinity gives torus m = 1 gives Moebius Band > m = 2 gives Square tube > m = 6 gives 12-sided polygon tube > m -> Infinity gives torus What are the independent variables? th and k are the independent parameters in a toroidal co-ordinate > system. Independent variables we call as parameters now. th is the polar co-ordinate/paarmeter and k the latitude of the torus > (similar to spherical cordinates) coordinate/parameter. Don't you think it would be better to switch to standard notation? This IS the standard notation, classical from earlier times. Notation changes over the years. Modern notation is clearer. And, Mathematica notation is not the same as mathematical notation. Here is what I think you are saying: g: {(t,k,m)| 0 <= t <= 4 Pi, 1 <= k <= 2m, 1 <= m <= oo} -> R^3 g(t,k,m) := ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), ( cos(t/2 + k Pi/m) + 3 ) sin(t), sin(t/2 + k Pi/m ) ), f_m(t,k) := g(t,k,m). Then for any m, 1 <= m <= oo, the image of f_m is a surface. And, f_1 is a MB, and f_oo is a torus. This is not a homeomorphism between a MB and a torus. If we restrict m to be in some finite interval, then we get a homotopy. For example, g gives a homotopy between the maps f_1 and f_2. However, two surfaces can be homotopic without being homeomorphic. And, you can't let m be oo if you are constructing a homotopy. A homeomorphism is different. Let M be a subset of R^3 that is a MB. Let T be a subset of R^3 that is a torus. Then a homeomorphism is a map h from M to T such that h is bijective, continuous, and its inverse is continuous. -- Marcus === Subject: Re: Moebius Band is not homeomorphic with a Torus > I have shown the images below and the parameterization / formula (but > held back nothing by way of stating something if at all I could ! ) > Frankly, I do not attach any significance to what type of (my ?!) > homeomorphism it is, as long as it is a valid one.The surface is just > twisting, bending and stretching of a membrane to a slightly altered > position.Just like twisting a long quadrilateral to new helical > position. If had known I would mention.I first thought it may be > bijective homeomorphism described by two continuous surface functions > for each topography or homotopy of sorts...It may not be so correct a > description with my scant familiarity with topology and skepticism ... http://img79.imageshack.us/img79/8162/mbtorussl6.jpg The top-left is a MB. The top-right is another surface (with two corners touching). The bottom-left has the two together. The bottom-right looks like the same as the bottom-left from another angle. So, what does this show? -- Marcus === Subject: Re: Moebius Band is not homeomorphic with a Torus Refers to the following earlier link : http://www.geocities.com/glnarasimham/MoebiusTorusMorph/TorusMoebiusMorph.ht m Narasimham === Subject: Re: notation for natural numbers > I've got books that do it both ways. Also, many books use bold (instead > of blackboard bold) for N, Z, Q, and R. OK, so name a book that does it both ways! (It's OK, I'm being > facetious.) Sorry! I realized after I posted it that it came out funny. -- Marcus === Subject: Re: JSH: Where in the world is Carmen San Diego ? >> You people are dumb as rocks. >> Factored anything lately, James? > The Harris trivial prime factoring function, TPFF(p), optimizes the time > neccesary to factorize any prime via surrogate quantisation and base-2 > compression technologies which are not implemented in the development > whatsoever. Surrogate factoring a consequence of Axiom of Choice ? Colloqium at 7:00pm > in G130 at Darden Hall, keynote address by JSH. > The hall is already sold out, I tried to get tickets. > Perhaps someone is reselling their ticket on ebay. > I'll go look. James' most admirable attribute is his relentless tenacity. Math and science, all knowledge, is the cumulative result of a half million years of arguing, squabbling, and general disagreement over what constiutes fact. Without such a process all knowledge would quickly evaporate due to lack of interest. I think that it's better to bullheaded and wrong than to be completely disinterested. === Subject: Re: JSH: Where in the world is Carmen San Diego ? <455e4e02$0$97260$892e7fe2@authen.yellow.readfreenews.net> <9JadndhLAtW0_sLYnZ2dnUVZ_o6dnZ2d@comcast.com> You people are dumb as rocks. >> Factored anything lately, James? > The Harris trivial prime factoring function, TPFF(p), optimizes the > time > neccesary to factorize any prime via surrogate quantisation and base-2 > compression technologies which are not implemented in the development > whatsoever. Surrogate factoring a consequence of Axiom of Choice ? Colloqium at > 7:00pm > in G130 at Darden Hall, keynote address by JSH. > The hall is already sold out, I tried to get tickets. > Perhaps someone is reselling their ticket on ebay. > I'll go look. > James' most admirable attribute is his relentless tenacity. Math and > science, all knowledge, is the cumulative result of a half million years of > arguing, squabbling, and general disagreement over what constiutes fact. > Without such a process all knowledge would quickly evaporate due to lack of > interest. I think that it's better to bullheaded and wrong than to be completely > disinterested. Ignorance is only skin deep but stupidity goes to the bone. === Subject: quadratic variation of log price process If I need to prove [logS logS] = int(signma^2(u)du) Then where should i exactly start from........ I know the distribution is lognormal for the price............. I know the process is ito..................[implies its brownian] Using Ito-isommetry i get the result............. Am i going wrong somewhere or can i skip a few steps... === Subject: Re: The truth about Set theory <87y7qcg5u6.fsf@phiwumbda.org> <87y7qczpwk.fsf@phiwumbda.org> <874pszklru.fsf@phiwumbda.org> <87lkmakfoz.fsf@phiwumbda.org This is not a complete answer. For one thing, mathematics contains > ideal elements, such as points. There is nothing in the external world > which can serve the role of a object with no length, breadth or width, The use of object here is very misleading. There is no law requiring everything to be a discrete object. It's not called the CONTINUUM for NOTHING, you know. Now that we know that most objects that are made of matter are basically quantized and NOT continuous, representing a line as (e.g.) a physical piece of string does not really seem accurate, but my point is, we still refer to the SPACE-TIME continuum. The real world has real space; real objects occur in it. There are REAL concrete points in space, AND in time. There is no obvious reason why elements of uncountable dense sets couldn't be modeled by THOSE points. More to the point, though, there are finitary geometries. What point really means has nothing to do with any sort of idealism or smallness, and everything to do with what axioms are being used to define points. Just as ABO himself has pointed out that a great many things about arithmetic (and about the numbers that arithmetic is about) can be derived withOUT assuming all of the usual axioms, it is also the case that a lot of things about points will hold in much smaller universes or models than the infinite/affine/ Euclidean one. Models in which you really CAN model points as discrete objects. > but there are objects which approximate it, and this still gives points > (and lines, etc.) a grounding in the world. That doesn't count. The grounding comes from things that DON'T merely approximate, but REALLY DO match, LIKE points in space. === Subject: Re: Cantor Confusion Take the set of natural numbers in form of a list or matrix: 1 > 11 > 111 > ... This matrix has length omega and width omega. And its diagonal has > length omega. No line has length omega. Therefore the width is larger > than any line. And the diagonal is longer than any line. This is > impossible. No, that is very possible. If you assert that there is no line longer than the diagonal, you have > good reasons, which can be proved. If you assert that the diagonal can be longer than any line, then you > have no reasons, because the diagonal consists of line elements and > cannot be where no line is. So your second assertion is outside of > logic and outside of any mathematics. Therefore I am not willing to > discuss this topic further. Does that mean you aren't going to post any more? > The correct result: There must be at least one line which is exactly as > long as the diagonal. There must be an infinite natural number. That is > impossible. Therefore, there is no actually infinite number of natural > numbers, -- Marcus === Subject: Re: Cantor Confusion ... > In > principle no axiom is necessary. But you need a few to have some start > to work with. That's the question. By means of axioms you can produce conditional > truth at most. I am interested in absolute truth. Axioms will not help > us to find it. I don't think we need any axioms. If you want to find absolute truth you should not look at mathematics. Really? There are two groups of order 4; could any truth be more > absolute than that? I think it depends on what the words mean. If the axioms are correct in your model, then the theorems are correct in your model. However, this seems to only be relative truth, i.e., if the axioms are absolutely true, then the theorem is absolutely true. Why did you pick the statement you did, rather than something like 2 + 2 = 4? -- Marcus === Subject: Re: Cantor Confusion I used this suitable word because it allows to speak of lines, columns > and diagonal. _Define_ what you mean! That would make it too simple! > I suspect you are speaking of matrices, lines, > columns and the like in default of a reasonable point of view. If you don't like it, say triangle or structure. I > propose to use infinite triangle in order to be clear and to show > that your commentary below fails to show anything. Instead of square > we should speak of equilateral. So we have an Equilateral Infinite > Triangle: EIT. Another misnomer. Cf. my posting in reply to Marcus. > <45550ba7$0$97245$892e7fe2@authen.yellow.readfreenews.net> At least I defined what I meant. And, I never said equilateral! -- Marcus === Subject: Re: Cantor Confusion >> >> I used this suitable word because it allows to speak of lines, >> columns and diagonal. >> >> _Define_ what you mean! That would make it too simple! LOL. Indeed. >> I suspect you are speaking of matrices, lines, >> columns and the like in default of a reasonable point of view. >> >> If you don't like it, say triangle or structure. I >> propose to use infinite triangle in order to be clear and to show >> that your commentary below fails to show anything. Instead of >> square we should speak of equilateral. So we have an >> Equilateral Infinite Triangle: EIT. >> >> Another misnomer. Cf. my posting in reply to Marcus. >> <45550ba7$0$97245$892e7fe2@authen.yellow.readfreenews.net > At least I defined what I meant. And, I never said equilateral! I prefered monangle or simply angle. F. N. -- xyz === Subject: Re: Cantor Confusion > Let he original matrix be A. >> [...] >> 1 >> 12 >> 123 >> ... >> Something is missing here: >> 1uuu... >> 12uu... >> 123u... >> ... No. Wrong. You have been discussing matrices. At least William Hughes did. > Are you both writing at cross purposes? WM is always writing at cross purposes to everyone! -- Marcus === Subject: Re: Cantor Confusion [...] >> >> Wrong. You have been discussing matrices. At least William Hughes >> did. Are you both writing at cross purposes? WM is always writing at cross purposes to everyone! I don't take him too seriously as long as it is fun. F. N. -- xyz === Subject: Re: Cantor Confusion > I'm saying that you don't understand what a mathematical definition is > but nonetheless want to pretend you do. If a mathematical definition > were just an abbreviation as you claim you wouldn't have any way to > tell one mathematical definition from another. Why not? Suppose I make the following definitions. Let N denote the set of natural numbers. Let R denote the set of real numbers. Then I can tell N and R are different because their defintions are different. If I write 0.5 is not in N, then this means the same as 0.5 is not in the set of natural numbers. And, it means something different from 0.5 is not in R. -- Marcus === Subject: Re: Cantor Confusion > What is it modern zen mathematikers do instead of thinking about the > truth of what they say? Sit around all day massaging their middle > legs? I mean really what is it they expect they get paid for? Proving theorems, of course. Fess up: you really knew that, didn't you? -- Marcus === Subject: Re: Cantor Confusion On 17 Nov 2006 14:49:13 -0800, R Tribble is a core of mathematics which every such civilization will accept. > > without axioms, yes. For instance: I + I = II (after translating + > and =). Therefore I call this an absolute truth. > > Which axioms are you using to describe the + and = operators? > >> None. Then how can you know it is true? An interesting question at least for a change. Problem is you don't advance knowledge of what's true by canonizing assumptions of truth in axioms. To find truth you need to find mechanically irreducible principles of what is universally false and truth is what's left over. I approach the problem in mechanical terms tautologically, by observing only differences can be true because there can be nothing different from differences. The idea of different from differences or the contradiction of contradiction is self contradictory which I take to be false in exhaustive mechanical terms. Consequently what's left over in tautological terms is contradiction or differences so that must perforce be true and be true of everything in universal terms. >In fact, it looks wrong to me. I think it should be I + I = I. But >perhaps I have a different implicit understanding of + and =. >What do your + and = mean? Well if you approach the problem from the perspective of finite tautological regression to self contradictory alternatives then = and + and all other things simply represent various compoundings of differences. = doesn't just apply to everything willy-nilly. Does the first = the second? No of course not. = and + do not apply to ordinality they only apply to cardinality where differences between intervals are the same. In other words + = - - under certain mechanical restrictions. The Peano axioms and the suc( ) axiom simply assume + as the foundation of mathematics when in point of fact - represents the foundation of all things in universal mechanical terms including mathematics. Hell the Peano axioms and the suc( ) axiom can't even produce straight lines. At best they produce various straight line segments of equal unit length but there is no guarantee that those straight line segments align with one another on any straight line. Only Newton's differential calculus can produce straight lines through his method of drawing tangents to curves. And those straight lines can be further subdivided in equal terms through infinitesimal subdivision and conversely integrated. That's where we get our knowledge of what's true mathematically when it comes to the manipulation of differences in terms of one another. All we need to do is recognize the various kinds of differences involved and the properties of the various kinds of differences such as unequal differences or ordinality and equal differences or cardinality which determine the properties of operations on them such as commutivity and distributivity for mathematical purposes. In other words none of these things require mathematical canonization in the form of axioms and all are demonstrable in mechanical terms. Technically (there you go, Brian) the only assumption I make is that self contradiction is false. And everything else is derived from that. Otherwise we might just say something like the self contradictory set is necessarily and universally empty and must necessarily remain so. However I think the point is nugatory regardless and we can be permitted the minor terminological assumption that self contradiction means false in every language possible. ~v~~ === Subject: Re: Cantor Confusion > Don't know what you mean by actual 'truth'. Can you give an example? Oh dear, oh dear, you have not, I detect, being paying attention. Probably not. I've been skipping over most of Lester's posts, since they haven't been too interesting. > Lester tells us bits of Truth all the Time, in particular our old > favourite ~v~~. This is Truth, baby, pure, 100% fat-free Truth. (I > mean, you can read it, right? Not-or-not-not. Oh. So that's what that is. > Just search the archives for Lester's explanation, Ah, so much to read, so little time. > and no, don't ask me, or anyone > but Lester, not that that is likely to get you any further than it got > everyone else.) I'm sure you are correct. > Zick: > go yourself. Ah, well. It seems you aren't interested in really learning anything, > after all. Hmm, perhaps you are catching on, after all. Despite lofty assertions, > it seems Lester's ultimate regression is always to foul language. Deep-seated anger? -- Marcus === Subject: Re: Cantor Confusion > The usefulness to other fields is demonstrated via the > scientific method, not by mathematical proof. You mean the empirical method not the scientific method. I think I meant what I said. > So mathematical axioms are to be empirically demonstrated now? I didn't say mathematical axioms are to be empirically demonstrated. I said that other fields demonstrate that mathematics is useful to them using the methods appropriate to those fields. > Are you saying you don't know what the word > proof means in mathematics? I'm saying you can't prove the truth of whatever you say in or about > mathematics. But, do you know what the word prove means in Mathematics? It isn't the same as what it means in English. > A major purpose of axioms is to avoid ambiguity. The main purpose of axioms is to provide assumptions of truth without > proof. Why do you think that? >> Are you going to illustrate the existence of infinites by production >> of one or more; or are you going to demonstrate the truth of their >> existence by some alternative means? You posit certain properties and >> characteristics of things you call infinites but don't show they can >> actually be realized in combination with one another. Sorry. Don't know what you mean. In particular, I don't know what you >mean by illustrate the existence, demonstrate the truth of their >existence, actually be realized. Can you give an example? I should give you examples of the examples of infinites I asked you > for? I didn't say that. I said you should give an example to show the meaning of the phrases I quoted. Pick something and illustrate its existence. > All you do in modern math is prove theorems from assumptions of truth. > Not exactly overtaxing intellectually but there it is. If by assumptions of truth you mean axioms, then that is correct. >Have you read any math books at the junior/senior college level or >above? Apparently more than you. Could be. Which math books have you read? >> Now I don't say there aren't cranks out there but there is also >> truth out there and you don't have a clue as to how to get at it. Why do you think truth is relevant to mathematics? I gave you the citation. I can't imagine why you think truth isn't > relevant to mathematics. What is the citation? I think I missed it. >Anyone who learns math is welcome to call themselves a mathematician. In other words anyone who learns to agree with you is welcome to call > themselves mathematicians? I didn't say that. >how it could be that math could be used in science of all types if the >mathematicians are really as you portray. Lucky guesses. I never said modern mathematikers, quantum empirics, > and relativists weren't lucky just that they were too lazy or stupid > to figure out the truth of what they were saying. We should all switch from mathematics to playing the lottery. > It isn't enough to learn the word. First, you have to > understand the concept. This takes work. But you said mathematical definitions are only abbreviations and now > alluva sudden you expect people to learn concepts instead? How droll. Sure. There is a difference between a thing and its name. Simply being able to recite a definition doesn't mean you understand it and can use it properly. >What books on the topic have you read? What courses have you taken? Apparently more than you. Could be. Which books have you read and which courses have you taken? > I can't even get you to discuss the truth of what you say. All I hear > about are your assumptions of truth in modern math. I never said that. -- Marcus === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The natural numbers count themselves. Bijection of initial segments of > column and lines 1 > 2 > 3 > ... > n <--> 1,2,3,...n If there is no infinite number then there are not infinitely many > numbers. This is clearly the point of contention. Consider, N, the set of all natural numbers. > By definition N only contains natural numbers. Cases > i: There is a largest natural number,n_L, then N={1,2,3,...,n_L} > In this case there are n_L natural numbers. Deleted. ii: There is no largest natural number. We will > write this as N={1,2,3,...} (the ... represent only natural > numbers). Set N has infinitely many > elements. Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. No case ii is there is no largest natural number. > and the set of natural numbers has infinite size. > (i.e.the natural numbers are counted by omega). You wish to distinguish there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) from case iii there is no largest natural number. > there are a finite number of natural numbers I don't see the distinction (if a set cannot be counted by an > infinite number, what else could it be but finite?) You are so much caught in set theory that you cannot even imagine that there are no infinite numbers, that some sets cannot be counted? >but > let us use the name case iv for there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) (it is really not a subcase of case ii) > Your case ii is the only case which considers an infinite set of natural numbers. This infinite set can be potentially infinite are actually infinite. Therefore I chose iia and iib. Edward Nelson, Princeton, says: There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. We shall not be far wrong if we call these the Platonic (P) and the Aristotelian (A) ways. But if you like to say ii and iv, I will accept it. OK. Knock yourself out. Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. As I remarked already, it does not say that the set has an ordinal number. > I don't use case iii. However, you still have to assume case ii. Starting from the axioms I have to assume case iv. Case ii then may be derived. I use the fact that the diagonal (=bijection, > d_nn) cannot be longer than every line, because it consists of what you > have called the line indexes. [Actually the lines consist of column indexes, it is the columns > which consist of line indexes. Since both sets consist of > exactly the natural numbers it doesn't really matter] I understood you saying that for 1,2,3,...,n being a line, n is the line index. The diagonal contains every column index. No line contains > every column index. Therefore the diagonal is longer > than every line. That assertion is wrong. You cannot show any d_nn the n of which is not in a line. Recall, we have assumed case ii, so we > have assumed no largest natural number. > In particular we have no last line. Correct. And we have no complete diagonal. And we have no complete column. > It is trivial to see that if there is no last line then > every line is shorter than the diagonal. > How can you claim that the diagonal cannot be longer > than every line? If you claim that, then you should name an element d_nn for which there is no n in any line. As you cannot show it, the diagonal cannot be longer than every line. So we have two results: 1) The diagonal must be longer than every line. 2) The diagonal cannot be longer than every line. This implies: The diagonal cannot be complete at all. The axiom of infinity is in contradiction with mathematics. For every element of the diagonal we must > have a line index and, hence, a line. This must hold in the extended > version too. No Recall. The extended version involves adding elements (not lines > and columns). Adding one element to every column adds a line, Only if the complete column does exist. But just that is wrong. > adding one element to every line does not add a column. > So if we start with the same number of lines and columns > we do not end with the same number of lines and columns. That should show you that the assertion of an actually infinite set of finite numbers is a self contradiction. This is my point of departure. If you say that the > diagonal can be longer than every line > then you say that there are > more natural numbers (elements of the bijection d_nn) than natural > numbers (indexes n). No. > Yes. d_nn are numbers. > I say that the diagonal consists of the sequence of all natural > numbers. > This means that for any natural number n the diagonal is longer > than {1,2,3,...,n}. Since every line can be written in the > form {1,2,3,...,n} this means that the diagonal is longer than > every line. You say the diagonal has order type omega. No line has order type omega. This prevents a bijection between lines and diagonal. We have an infinite number of lines. We say so. But we forget that this implies to have an infinite line (in case ii). > By case ii there > is no line with infinite index. Thus there is no element > of the diagonal with infinite index. So the set > of elements of the diaongal d_nn is exaclty the > set of natural numbers < omega. > Why does addition of one element yield different results for columns, diagonal and lines? The number of lines and number of columns is the same. Why does addition of one element yield different results for columns, diagonal and lines? > However, the set of initial segments in the first colum > does not correspond to the set of lines. It does (the set of finite segments). Look at the introductory sketch. > There is > one initial segment {1,2,3,...} that is in the first column > but is not in any line. The existence of this initial segment > means that adding a element to the first column is > different than adding an element to any line. > However, the existence of the initial segment {1,2,3,...} > does not change the number of lines, because the > initial segment {1,2,3,...} is no associated with any line. Correct. Therefore there is no bijection between lines and initial segments of columns. Therefore, there is no infinite number omega of lines. > So the existence of the initial segment {1,2,3,...} cannot > be invoked to claim there is no bijection between > the lines and the columns. > The bijection is only valid for finitely many segments (finitely many numbers) and lines with finite index (finite numbers). We cannot give an upper bound, but we can exclude an infinite ordinal and cardinal. === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > The natural numbers count themselves. Bijection of initial segments of > column and lines 1 > 2 > 3 > ... > n <--> 1,2,3,...n If there is no infinite number then there are not infinitely many > numbers. This is clearly the point of contention. Consider, N, the set of all natural numbers. > By definition N only contains natural numbers. Cases > i: There is a largest natural number,n_L, then N={1,2,3,...,n_L} > In this case there are n_L natural numbers. Deleted. ii: There is no largest natural number. We will > write this as N={1,2,3,...} (the ... represent only natural > numbers). Set N has infinitely many > elements. Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. No case ii is there is no largest natural number. > and the set of natural numbers has infinite size. > (i.e.the natural numbers are counted by omega). You wish to distinguish there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) from case iii there is no largest natural number. > there are a finite number of natural numbers I don't see the distinction (if a set cannot be counted by an > infinite number, what else could it be but finite?) You are so much caught in set theory that you cannot even imagine that > there are no infinite numbers, that some sets cannot be counted? but > let us use the name case iv for there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) (it is really not a subcase of case ii) Your case ii is the only case which considers an infinite set of > natural numbers. This infinite set can be potentially infinite are > actually infinite. Therefore I chose iia and iib. Edward Nelson, > Princeton, says: There are at least two different ways of looking at > the numbers: as a completed infinity and as an incomplete infinity. We > shall not be far wrong if we call these the Platonic (P) and the > Aristotelian (A) ways. But if you like to say ii and iv, I will accept > it. OK. Knock yourself out. Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. As I remarked already, it does not say that the set has an ordinal > number. I don't use case iii. However, you still have to assume case ii. Starting from the axioms I have to assume case iv. Case ii then may be > derived. No. You are trying to show that assuming case iia leads to a contradiction. To do this you need to assume case iia. In particular you are assuming - the set, N, of all natural numbers exists - the set N is infinite. - N has no last element I use the fact that the diagonal (=bijection, > d_nn) cannot be longer than every line, because it consists of what you > have called the line indexes. [Actually the lines consist of column indexes, it is the columns > which consist of line indexes. Since both sets consist of > exactly the natural numbers it doesn't really matter] I understood you saying that for 1,2,3,...,n being a line, n is the > line index. The nth line is composed of the column indexes 1,2,3,...,n The diagonal contains every column index. No line contains > every column index. Therefore the diagonal is longer > than every line. That assertion is wrong. You cannot show any d_nn the n of which is not > in a line. The diagonal contains all the d_nn. No line contains all the d_nn. Therefore the diagonal is longer than every line. Recall, we have assumed case ii, so we > have assumed no largest natural number. > In particular we have no last line. Correct. And we have no complete diagonal. And we have no complete > column. We have assumed case iia. An infinite set with no last element exists. It is trivial to see that if there is no last line then > every line is shorter than the diagonal. > How can you claim that the diagonal cannot be longer > than every line? If you claim that, then you should name an element d_nn for which there > is no n in any line. The set of all lines contains every d_nn. No single line contains every d_nn. The diagonal contains all d_nn. The diagonal is longer than every line. > So we have two results: > 1) The diagonal must be longer than every line. > 2) The diagonal cannot be longer than every line. No. The diagonal contains all the d_nn. No line contains all the d_nn. The diagonal is longer than every line. > This implies: The diagonal cannot be complete at all. The axiom of > infinity is in contradiction with mathematics. For every element of the diagonal we must > have a line index and, hence, a line. This must hold in the extended > version too. No Recall. The extended version involves adding elements (not lines > and columns). Adding one element to every column adds a line, Only if the complete column does exist. But just that is wrong. > If we assume case iia the complete column does exist.. > adding one element to every line does not add a column. > So if we start with the same number of lines and columns > we do not end with the same number of lines and columns. That should show you that the assertion of an actually infinite set of > finite numbers is a self contradiction. No. it just shows that the assertion of an actually infinite set of finite numbers leads to results that you do not like. However, these are not contradictory results. This is my point of departure. If you say that the > diagonal can be longer than every line > then you say that there are > more natural numbers (elements of the bijection d_nn) than natural > numbers (indexes n). No. Yes. d_nn are numbers. I say that the diagonal consists of the sequence of all natural > numbers. > This means that for any natural number n the diagonal is longer > than {1,2,3,...,n}. Since every line can be written in the > form {1,2,3,...,n} this means that the diagonal is longer than > every line. You say the diagonal has order type omega. No line has order type > omega. This prevents a bijection between lines and diagonal. By case ii there is an infinite set (a set of order type omega) that only contains finite elements. The lines are the elements of the set of lines. The fact that no line has order type omega does not mean that the set of lines does not have order type omega. Both the diagonal and the set of lines have order type omega. We have an infinite number of lines. We say so. But we forget that this implies to have an infinite line (in > case ii). No case iia states exactly the opposite. We have an infinite set (the natural numbers) that does not have an infinite element. By case ii there > is no line with infinite index. Thus there is no element > of the diagonal with infinite index. So the set > of elements of the diaongal d_nn is exaclty the > set of natural numbers < omega. Why does addition of one element yield different results for columns, > diagonal and lines? The columns and the diagonal both contain an infinite initial segment. No line contains an infinite initial segment. The number of lines and number of columns is the same. Why does addition of one element yield different results for columns, > diagonal and lines? The columns and the diagonal both contain an infinite initial segment. No line contains an infinite initial segment. > However, the set of initial segments in the first colum > does not correspond to the set of lines. It does (the set of finite segments). Look at the introductory sketch. > The set of finite segments is the same. There is one segment that is not finite. The set of initial segments does not correspond to the set of lines. > There is > one initial segment {1,2,3,...} that is in the first column > but is not in any line. The existence of this initial segment > means that adding a element to the first column is > different than adding an element to any line. > However, the existence of the initial segment {1,2,3,...} > does not change the number of lines, because the > initial segment {1,2,3,...} is no associated with any line. Correct. Therefore there is no bijection between lines and initial > segments of columns. Therefore, there is no infinite number omega of > lines. > There is an infinite number of finite initial segments. The set of finite initial segments is the set N. By iia the set N is infinite. There is a bijection between the lines and the finite initial segments. Therefore there is an infinite number of lines. > So the existence of the initial segment {1,2,3,...} cannot > be invoked to claim there is no bijection between > the lines and the columns. The bijection is only valid for finitely many segments (finitely many > numbers) No. The bijection is only valid for finite (not finitely many) segments. By iia there are an infinitely many finite sements. > and lines with finite index (finite numbers). By iia there are an infinite number of finite numbers. >We cannot give > an upper bound, but we can exclude an infinite ordinal and cardinal. > Not if you assume the set of all natural numbers, N, exists. -William Hughes === Subject: Re: Cantor Confusion > On Thu, 16 Nov 2006 02:02:49 -0500, Marcus >> Provability of what pray tell? If you're not concerned with proving >> the truth of what you say in mathematics exactly when are you not >> discussing philosophy every time you say anything in mathematics? Do you really not know the mathematical meaning of the word prove? If >so, I (and others) could try to explain it to you. But, if you are just >being argumentative, we won't bother. What is it you think you're proving? Does that mean you don't know the mathematical meaning of the word prove? It isn't the same as the English meaning. >Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. Are you sure that is what Stephen thinks? A set isn't a proper subset of itself. If we have two sets A and B, then we say that A is a proper subset of B if every element in A is also in B and there is some element in B that is not in A If we take B = A, then every element in B is also in A. So, A is not a proper subset of itself. -- Marcus === Subject: Re: Cantor Confusion >> On Thu, 16 Nov 2006 02:02:49 -0500, Marcus > Provability of what pray tell? If you're not concerned with proving > the truth of what you say in mathematics exactly when are you not > discussing philosophy every time you say anything in mathematics? >>Do you really not know the mathematical meaning of the word prove? If >>so, I (and others) could try to explain it to you. But, if you are just >>being argumentative, we won't bother. >> >> What is it you think you're proving? > Does that mean you don't know the mathematical meaning of the word > prove? It isn't the same as the English meaning. >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his standard pattern. He really is pathetic. It is amusing to see how increasingly pathetic he becomes. Stephen === Subject: Re: Cantor Confusion >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his > standard pattern. He really is pathetic. It is amusing to see > how increasingly pathetic he becomes. And, such a silly lie. What could he hope to gain? -- Marcus === Subject: Re: Cantor Confusion >> On Thu, 16 Nov 2006 02:02:49 -0500, Marcus >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his > standard pattern. He really is pathetic. It is amusing to see > how increasingly pathetic he becomes. And, such a silly lie. What could he hope to gain? Attention! === Subject: Re: Cantor Confusion >Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. >> >> Are you sure that is what Stephen thinks? >> >> I see Lester has resorted to lying. This is all part of his >> standard pattern. He really is pathetic. It is amusing to see >> how increasingly pathetic he becomes. > And, such a silly lie. What could he hope to gain? I am not sure if it is really even just a lie. Honestly I do not think Lester is capable of comprehending anything mathematical, so when he tries to restate what anybody else has said he gets it wrong not just because of dishonesty (he has clearly demonstrated a decided dishonest streak), but because of ineptitude. When he reads something like A set is infinite if there exists a bijection between itself and a proper subset. all that comes through is * set is infinite ** ***** ****** * ********* ****** itself *** * proper subset. and that is somehow boiled down into An infinite set is a proper subset of itself. It reminds me of this: http://www.bakbone.com/newsletter/images/ginger_large.gif Stephen === Subject: Re: Cantor Confusion Crank means someone who makes up pejorative names for people whose > language they do not understand and thinks that definitions which are > explained in many books are private. So you are a crank? I'll leave this to others to judge. > 1) You call others cranks, whose language you don't understand. Perhaps. > 2) You do not understand words used by many current text books. Very true. Although, not true of the quotes you gave from a set theory text book. > BTW: Do you really think it is a proof of your superior intellect if > every second word of yours is sorry don't know? Yes. A wise man knows what he doesn't know. > How can you do mathematics without axioms? A major purpose of axioms is > to avoid ambiguity. How has it been done over 4000 years? Since Euclid (and presumably for some time before), much of it has been done using axioms. As time has gone by, more and more of mathematics has become axiomatized. > I can verify I'm using words with the same meanings as other people by > asking them what definitions they are using, then seeing if they are the > same as mine. How could you see that if you don't know what the words in their > definitions mean. An interesting question. It doesn't really pose a problem in practice, but I don't think I can give a short answer. > We use technical terms to refer to precisely defined mathematical > concepts. Why do you say we if you talk about mathematicians? What do you mean? -- Marcus === Subject: Re: Cantor Confusion The fact that something is true for all sets of the form > {1,2,3,...n} where n is a finite natural number, > does not mean that it is true for N. Oh yes, exactly that it means, because N consists of nothing else than > natural numbers. There are no ghosts in mathematics. How do you know this? Do you have any sort of rationale or proof? It > seems such a silly thing to say. Consider: Each of the following sequences has a last element: 1 > 1 2 > 1 2 3 > 1 2 3 4 > 1 2 3 4 5 > ... This sequence does not have a last element: 1 2 3 4 5 ... This last sequence has three dots on the right. None of the other > sequences do. So, this last sequence is clearly different in some way > from all the other sequences. Yes. We do not know its last element. Perhaps it can change its size. > But mathematics does not obey commands. Neither interpreted as a > command nor as a magic formula the ... can create infinity. So, let's try a slightly different one: We know the last element of each of the following sequences: 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 ... We don't know the last element of this sequence: 1 2 3 4 5 ... Therefore, there are things that are true for 1,...,n for all n in N that are not true for N. But, I believe you said that if something is true for 1,...,n for all n in N then it is true for N. -- Marcus === Subject: Re: Cantor Confusion > On Thu, 16 Nov 2006 01:35:12 -0500, Marcus > >> It is difficult to answer this question, because the expression set >> is occupied in modern mathematics by collections of elements which are >> actually there (you don't know what that means, imagine just a set as >> you know it). Such infinite sets do not exist. >> >> While infinite collections in any physical sense are not possible, why >> are imaginary infinities, such as sets of numbers must be, unimaginable? Why are square circles unimaginable? Depends on what you mean by unimaginable. >For that matter, we can always switch from Platonism to formalism and >declare the question of whether sets really exist to be a philosophical >question. So is the switch from platonism to formalism a philosophical question? Yes. Platonism and formalism are philosophies of mathematics. Regardless of which you prefer (or if you prefer something else), it doesn't change which theorems are provable in which axiom systems. -- Marcus === Subject: Re: Cantor Confusion Indeed. If people *object* to an axiom, that is philosophy. But if people choose a set of axioms, that is what? Everyone is welcome to choose their own axioms. That's mathematics? Of course. And if people not decide to use an axiom, that is what? > But if people decide not to use an axiom, that is philosophy? I said, If people *object* to an axiom, that is philosophy. If you want to see what theorems you can prove using a particular set of axioms, that is mathematics. Whether your results will be interesting or useful to other people is a different question. > Would like to do. Please le me know which words are available in your > universe of discourse. I told you several times that the terminology in any modern textbook is > fine. For some reason you do not like this answer. I told you the terminology used in a modern textbook to show that > finished infinity is used there. For some reason you do not like to > understand it. Some mathematicians object to the Axiom of Infinity on the grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity. Are you intentionally being stupid? That quote doesn't use finished infinity nor does it give a definition of finished entity. The quote is simply a philosophical remark in a math textbook. > What would be something that is actually infinite? Read Cantor, he can explain it better than me. Hah! So, you can't give an example. Then please stop using the words! > e. An infinite number is a number other than the natural numbers. An infinite number would be a number other than a natural number. Are you agreeing or disagreeing? I am astonished that you cannot understand simplest sentences. I only seem to have this problem with your sentences. I asked whether you agreed with a statement and you replied by offering a different statement. Do you disagree with the original or agree? > You seem > to have difficulties with conditional constructs. Should you ever > intend to study mathematics be prepared that such constructs will > appear quite frequently. How would you know such constructs appear frequently if someone studies mathematics? What mathematics have you studied? Where did you study it? Do you have a doctorate in Mathematics? Who was your thesis advisor? > If an actually infinite set of numbers existed, and if neighbouring > elements had a fixed distance from each other, then the set must > contain an infinite number. Is that a no or a yes? Read again, simplified: If neighbouring elements have a fixed distance, > the answer is yes. > If neighbouring elements have not a fixed distance like the rational > numbers: the answer is no Let's try a simpler question: Does an actually infinite set exist? -- Marcus === Subject: Re: Cantor Confusion If an actually infinite set of numbers existed, and if neighbouring > elements had a fixed distance from each other, then the set must > contain an infinite number. Is that a no or a yes? Read again, simplified: If neighbouring elements have a fixed distance, > the answer is yes. > If neighbouring elements have not a fixed distance like the rational > numbers: the answer is no Let's try a simpler question: Does an actually infinite set exist? WM has already committed himself to a situation in which as set like the rationals need not contain any infinite element but that a proper subset of it, like the integral rationals, must contain an infinite element. One wonders how a proper subset can contain an element not in the containing superset? === Subject: Re: Cantor Confusion > Again, no answer. If you cannot understand this answer, then we should stop here. I assure you that no one is insisting that you continue to post. I think everyone will agree that what you have already posted adequately describes your position. -- Marcus === Subject: Re: Cantor Confusion >> {1,2,3} is the collection of, and a convenient expression to write >> {that we are talking about, the numbers 1 ,2, and 3. >> What are we talking about, when we write >> >> { } >> >> ? The same as when we write I would like to suggest the name finished emptiness for that. We could simply agree that all symbols, letters, numerals denote emptiness. This would mean that all posts in this thread are empty. Perhaps it would be a fitting end to the discussion. It seems rather perverse to insist that some symbols be replaced by blanks. We aren't allowed to use whatever symbols we want? -- Marcus === Subject: Re: Cantor Confusion > No. The hidden errors can better be recognized at the roots. Perhaps, but irrelevant. If you can't find the hidden errors in the > modern formulations, a likely explanation is that the hidden errors > have been removed in the process of changing the formulations. > Regardless, if the hidden errors are still there, your only hope of > convincing people is to point to them in the formulations that they > know. The most fundamental hidden error is that sets are described a > potentially infinite (going on for ever) but taken and treated as if > they were actually infinite. As I just said, you need to point to the errors in the formulations that people currently use. If the error is treating sets as if they are actually infinite, please state what the error is using modern terminology and/or give an example using modern terminology. -- Marcus === Subject: Re: Cantor Confusion In the below post, I was just trying to paraphrase what you are saying. Please don't try to paraphrase what I said, because I don't believe > that you understand it sufficiently. So in most cases you will fail to > repeat my ideas. Please quote only full sentences. Unfortunately, none of your full sentences make any sense. > I didn't say I would say that or that I understood what you were trying > to say. The question was whether you ever said that it. I hope this question > as been settled now. Are you intentionally being silly? You were saying that I had used the phrase myself, and so must know what it means. If I say, Peter says fiddlebok, does that mean I know what fiddlebok means? > In fact, I don't know what you you mean by the phrase. Did you There is no misunderstanding possible. You refuted Lester's > interpretation, by proposing to have a better one: That doesn't seem to be what WM is saying. He seems to be saying that > the notion of a completed infinity leads to either absurdities or > contradictions. Perhaps he thinks the way to avoid these absurdities is > to only consider things that can be physically produced. I now that cranks never admit having made an error. But do you think > that obvious lies like this are a way to reach your aim? What aim is that? > Why does it > make you believe that there is an infinite initial segment 1,2,3... ? Sorry. I don't understand. 1,2,3,... is the set of natural numbers. > forever, and I agreed to it. You seem to be asking me why I believe the > set of natural numbers goes on forever. But, we just agreed that was > true. So, what are you asking me? > If it only goes on forever without being completet anywhere, then there > is no chance to find out whether all natural numbers are sufficient to > enumerate all real numbers or not. > I don't know what you mean by completed anywhere. > The completed initial segment contains every natural number. > Another segment contains not every natural number. Let's recap. We agreed that the natural numbers go on forever. You then asked me why I believe that there is an infinite initial segment 1,2,3... I said I didn't understand the question: you seemed to be asking me why I believe the natural numbers go on forever (something we both agreed to). You now say that completed infinite initial segment means it contains every natural number. I still don't know what you mean. When I say natural numbers, I'm not excluding any natural numbers. And, the natural numbers go on forever. So, I still don't understand what you are asking. What else do you think I believe? -- Marcus === Subject: Re: Cantor Confusion > >> I see. But recently you used the word completed infinity. >> I don't think I ever said that. Do you have a quote? >> Here it is: >> In the below post, I was just trying to paraphrase what you are >> saying. Please don't try to paraphrase what I said, because I don't believe > that you understand it sufficiently. This is due to your conceptional weakness. [...] > >> I didn't say I would say that or that I understood what you were >> trying to say. The question was whether you ever said that it. I hope this > question as been settled now. > >> In fact, I don't know what you you mean by the phrase. Did you There is no misunderstanding possible. You refuted Lester's > interpretation, by proposing to have a better one: That doesn't seem to be what WM is saying. He seems to be saying that > the notion of a completed infinity leads to either absurdities or > contradictions. Perhaps he thinks the way to avoid these absurdities > is to only consider things that can be physically produced. I [k]now that cranks never admit having made an error. Introspection? But do you think that obvious lies like this are a way to reach your > aim? > I cannot spot any lie. -- Marcus === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? > Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal number. My case iia is in complete agreement with the axiom of infinity. I am in a hurry. Perhaps I will give some more comments later. Although the principle positions have been cleared. === Subject: Re: Cantor Confusion I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? It means that you in waters over your head and can't swim. Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal > number. Because ordinal number and cardinal number have not yet been given definitions. Those definitions, and others, follow from the axioms rather than preceding them. WM again has cart-before-horse-itis. === Subject: Re: Cantor Confusion > WM again has cart-before-horse-itis. He thinks he doesn't need a horse. -- Marcus === Subject: Re: Cantor Confusion > WM again has cart-before-horse-itis. He thinks he doesn't need a horse. He does need the other half to be a complete horse. === Subject: Re: Cantor Confusion I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? http://idioms.thefreedictionary.com/knock+yourself+out -- Marcus === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal > number. My case iia is in complete agreement with the axiom of > infinity. > The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements The axiom of infinity does not say anything about ordinal or cardinal numbers. However, given that the set N exists and the defnition of ordinal and cardinal numbers, it is easy to see that if N exists it must have both an ordinal and a cardinal number. Ordinal. Every natural number is an ordinal. Every initial sequence of ordinals is an ordinal. (an initial sequence is a set of ordinals, such that if a is in the set every ordinal less than a is in the set). The set of all natural numbers is an initial sequence of ordinals. Therefore the set of natural numbers is an ordinal. Cardinal. Cardinal numbers are equivalence classes of sets under the equivalence relation bijection. Since any set has a bijection to itself, every set must be in an equivalence class. So the set of natural numbers has a cardinal number.. - William Hughes === Subject: Re: Cantor Confusion If you want to find absolute truth you should not look at mathematics. >> Perhaps we should replace absolute truth with culturally neutral >> truth, or in other words, truth without any cultural, religious, or >> philosophical bias. [...] Thinking about this question >> leads most of us to believe that there is a core of mathematics which >> every such civilization will accept. > without axioms, yes. For instance: I + I = II (after translating + > and =). Therefore I call this an absolute truth. Which axioms are you using to describe the + and = operators? Axioms? For which purpose? Do you think the symbols constituting the words constituting the axioms are easier or clearer to understand than the symbols + and =? Take an apple and then another apple. Show the apples first apart and then together. Repeat with oranges or fingers or mixed objects, possibly. That defines all that is needed. === Subject: Re: Cantor Confusion without axioms, yes. For instance: I + I = II (after translating + > and =). Therefore I call this an absolute truth. Which axioms are you using to describe the + and = operators? Axioms? For which purpose? Perhaps definitions and axioms wold have been better. > Do you think the symbols constituting the > words constituting the axioms are easier or clearer to understand than > the symbols + and =? Take an apple and then another apple. Show > the apples first apart and then together. Repeat with oranges or > fingers or mixed objects, possibly. That defines all that is needed. No number of specific cases ever defines a general rule sufficiently for purposes of mathematics. === Subject: Re: Cantor Confusion <455c7752$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net >> http://mathworld.wolfram.com/InitialSegment.html >> What you call complete initial segment is not an *initial* segment >> but the whole set of lines. What is a name? You should rephrase what you mean. What *counts* is this: It is asserted that the number of natural > numbers is omega It is asserted that the *set* of natural numbers is (does exist) and > is named omega. The *cardinality* of omega is aleph_0. The cardinality of omega is omega. It is usual to denote it by aleph_0, but it is allowed to denote it by omega. Even Cantor is said to have done so ... (the elements of the first column = number of lines). The ordinal number of lines is omega. Its cardinal number is aleph_0. > Neither omega nor aleph_0 are /natural/ numbers. Neither omega nor > aleph_0 are elements of omega. Period. And there are not enough natural numbers that we could collect omega or aleph_0 of them, although this wrong definition exists. We see the lack by increasing the number of elements and the elements by 1. By adding 1 element to the first column we see that, if this assertion > is correct, the number increases to omega + 1. It depends on how you in the present case define adding 1 element to > the first column. As pointed out: L' := { 0, 1, 2, ..., x } is of ordinal type omega + 1 > L'' := { x, 0, 1, 2, ... } is (still) of ordinal type omega Besides of this | L' | = | L'' | = aleph_0 is valid. What L + 1 do you mean? L' or L''? But are there enough lines? There are as many lines as you have defined. No. How did you hit on that? That means: The finite natural numbers are not sufficient > to stand in bijection with all omega elements of the column, Here you are: B := { <0, 1>, <1, 2>, <2, 3>, ... } B is an explicit bijection between the naturals (elements of omega) and > the numbers in the first column. B does not include omega. If omega were only the fact that this bijection does include all natural numbers, then you had no problem with B. But if omega is considered a number which even can be increased, then the idea breaks down and you must assume that there are more natural numbers d_nn than natural numbers n. === Subject: Re: Cantor Confusion http://mathworld.wolfram.com/InitialSegment.html >> What you call complete initial segment is not an *initial* segment >> but the whole set of lines. What is a name? You should rephrase what you mean. What *counts* is this: It is asserted that the number of natural > numbers is omega It is asserted that the *set* of natural numbers is (does exist) and > is named omega. The *cardinality* of omega is aleph_0. The cardinality of omega is omega. It is usual to denote it by aleph_0, > but it is allowed to denote it by omega. Even Cantor is said to have > done so ... Since WM has so often claimed that Cantor is all wrong, he is hardly now in a position to justify anything by saying that Cantor was right about it. > And there are not enough natural numbers that we could collect omega or > aleph_0 of them, although this wrong definition exists. Since omega is, by definition, such a collection, WM is henceforth barred from using it by his own argument. > We see Who is this we that sees thing that are no there? B := { <0, 1>, <1, 2>, <2, 3>, ... } B is an explicit bijection between the naturals (elements of omega) and > the numbers in the first column. B does not include omega. But B is order isomorphic to omega. Which for ordinality is just as good. > If omega were only the fact that this > bijection does include all natural numbers, then you had no problem > with B. But if omega is considered a number which even can be > increased, Does WM claim that there are no ordinals of which omega is a proper subset (where it is understood that every ordinal is the set of all previous ordinals)? > you must assume that there are > more natural numbers d_nn than natural numbers n. Not unless your brain is cross wired. N need never be a proper subset, or proper superset, of N. === Subject: Re: Cantor Confusion > http://mathworld.wolfram.com/InitialSegment.html What you call complete initial segment is not an *initial* > segment but the whole set of lines. >> What is a name? >> You should rephrase what you mean. >> What *counts* is this: It is asserted that the number of natural >> numbers is omega >> It is asserted that the *set* of natural numbers is (does exist) >> and is named omega. The *cardinality* of omega is aleph_0. The cardinality of omega is omega. The cardinality of omega is |omega| not omega. > It is usual to denote it by aleph_0, but it is allowed to denote it by > omega. Being imprecise does not help to clarify things. > Even Cantor is said to have done so ... Irrelevant. >> (the elements of the first column = number of lines). >> The ordinal number of lines is omega. Its cardinal number is aleph_0. >> Neither omega nor aleph_0 are /natural/ numbers. Neither omega nor >> aleph_0 are elements of omega. Period. And there are not enough natural numbers that we could collect omega > or aleph_0 of them, In set theory there is no task to collect natural numbers. Set theory is not about processes of collecting numbers. > although this wrong definition exists. You have already been informed about this misconception of yours. > We see the lack by increasing the number of elements and the elements > by 1. Anything new? >> By adding 1 element to the first column we see that, if this >> assertion is correct, the number increases to omega + 1. >> It depends on how you in the present case define adding 1 element to >> the first column. As pointed out: >> L' := { 0, 1, 2, ..., x } is of ordinal type omega + 1 >> L'' := { x, 0, 1, 2, ... } is (still) of ordinal type omega >> Besides of this | L' | = | L'' | = aleph_0 is valid. >> What L + 1 do you mean? L' or L''? > As W. Hughes (?) has already pointed out: There is no list (in the original sense, i. e. function with domain omega) L' (i. e. having co-domain L'). >> But are there enough lines? >> There are as many lines as you have defined. >> No. >> How did you hit on that? >> That means: The finite natural numbers are not sufficient >> to stand in bijection with all omega elements of the column, >> Here you are: >> B := { <0, 1>, <1, 2>, <2, 3>, ... } >> B is an explicit bijection between the naturals (elements of omega) >> and the numbers in the first column. B does not include omega. You tend to repeat yourself. > If omega were only the fact that this bijection does include all > natural numbers, then you had no problem with B. I have no problem with B and this bijection does include all natural numbers. > But if omega is considered a number which even can be increased, At _your_ own risk you may consider omega to be anything you want. Mathematically a definition of increase would be preferred. > then the idea breaks down and you must assume that there > are more natural numbers d_nn than natural numbers n. Set theory stands like a rock undaunted by WM's delusions. F. N. -- xyz === Subject: Re: Cantor Confusion >> What *counts* is this: It is asserted that the number of natural >> numbers is omega >> It is asserted that the *set* of natural numbers is (does exist) >> and is named omega. The *cardinality* of omega is aleph_0. The cardinality of omega is omega. The cardinality of omega is |omega| not omega. Kunen's Set Theory defines |A| to be the least ordinal that can be bijected with A. So, with this definition, |omega| = omega. > although this wrong definition exists. You have already been informed about this misconception of yours. I fear that WM thinks he is informing us of our misconceptions. -- Marcus === Subject: Re: Cantor Confusion > What *counts* is this: It is asserted that the number of natural > numbers is omega It is asserted that the *set* of natural numbers is (does exist) > and is named omega. The *cardinality* of omega is aleph_0. >> >> The cardinality of omega is omega. >> >> The cardinality of omega is |omega| not omega. Kunen's Set Theory defines |A| to be the least ordinal that can be > bijected with A. So, with this definition, |omega| = omega. You are absolutely right. But we debate the wrong thing instead of | What counts is this: It is asserted that the number of natural | numbers is omega (the elements of the first column = number of lines). | By adding 1 element to the first column we see that, if this assertion | is correct, the number increases to omega + 1. But are there enough | lines? No. That means: The finite natural numbers are not sufficient | to stand in bijection with all omega elements of the column, but only | with the finite ones. Therefore the unavoidable conclusion is: There | are not omega finite numbers. `---- F. N. -- xyz === Subject: Re: Cantor Confusion > Indeed. If people *object* to an axiom, that is philosophy. >> But if people choose a set of axioms, that is what? >> Everyone is welcome to choose their own axioms. >> That's mathematics? >> Of course. And if people not decide to use an axiom, that is what? Mathematics obviously. says so. Any reasonable person would say so. It's mathematics on > a system which is missing that axiom. It's still perfectly good > mathematics. Haven't you seen people discuss ZF vs. ZFC for instance? Both are > mathematics. ZFC has the axiom of choice, people working in ZF > choose not to use that axiom. Haven't we seen people discuss the axiom of infinity? Hrbacek and Jech, for instance, report that. But some self proclaimed crank doesn't want to call that mathematics, because then he had to admit that something he did not know is mathematics nevertheless. That is why I asked the silly questions above. Of course all that is mathematics. === Subject: Re: Cantor Confusion > Of course all that is mathematics. > Much of what WM claims is mathematics is nonsense and other parts are just not mathematics, and very little of it actually is mathematics. === Subject: Re: Cantor Confusion idea in order to distinguish it from numbers which can be written in >lists and can be subject to a diagonal proof. I call only those >entities numbers which can be put in trichotomy with each other. I don't know what a diagonal proof and trichotomy may be. Excuse me. 1) The diagonal proof by Cantor shows that any list of real numbers is incomplete. For this purpose we use a list (= injective sequence) of real numbers and exchange the n-th digit of the n-th number. The changed digits put together yield a real number which differs from each list entry at least at one position (it differs at position n from the n-th list entry). This proof requires that all digits of numbers like sqrt(2) do exist and can be exchanged. That assumption is wrong. 2) The fact that for each pair of numbers a and b we have a < b or a = b or a > b is called Trichotomy. For such numbers as P = [pi*10^10^100] and Q = (the same number but the last digit exchanged by 5) the order by size cannot be determined. [x] denotes the integer part of x. === Subject: Re: Cantor Confusion > >Sqrt(2) does exist as the diagonal of the square. But I call that an >idea in order to distinguish it from numbers which can be written in >lists and can be subject to a diagonal proof. I call only those >entities numbers which can be put in trichotomy with each other. I don't know what a diagonal proof and trichotomy may be. Excuse me. > 1) The diagonal proof by Cantor shows that any list of real numbers is > incomplete. For this purpose we use a list (= injective sequence) of > real numbers and exchange the n-th digit of the n-th number. The > changed digits put together yield a real number which differs from each > list entry at least at one position (it differs at position n from the > n-th list entry). This proof requires that all digits of numbers like > sqrt(2) do exist and can be exchanged. That assumption is wrong. WRONG AGAIN! Cantor's diagonal proof merely requires that the nth number in the list have, in principle even if not in practice, a determinable nth digit, which is quite a different issue. For example, it is definitely the case that for any given position n, the nth digit of sqrt(2) is, at least in principle, determinable, however impractical it might be to carry out such a determination. So WM's assumption is not, in fact, required. And WM's objections are, in fact, nonsense. === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> numbers. What the lines contain (occupancy) does not effect the number of > sequence menbers. It does, if the members enumerate themselves. This is the case in the > present EIT: 1 > 12 > 123 > ... >> Equivocation: Adding one element names two different things >> (changing the occupancy vs. changing the domain). It names two different things only if there are two different things in > the initial bijection. But that cannot be the case because these things > are identical (except that one is noted vertically and the other one > horizontally.) Hence, any difference can only be that the nunmber of > these things is different. That is what I proved. I have yet to see any WM proof that is mathematically or logically > satisfactory. There are always hidden assumptions that are unwarranted. I use the hidden assumption that there are not more natural numbers d_nn than natural numbers n. If you can accept that the are more natural numbers than natural numbers, then we have different world views which do not fit together and cannot be united. So we should stop here. Your mathematics is not mine and vice versa. I assume that part of my mathematics belongs to absolute truth which does not rely on arbitrary axioms. Perhaps sometimes an alien civilization will be contacted. It would be interesting to get to know their mathematics. But the chances are very low. Anyhow it is not useful that we comment the contributions of each other any longer, because the positions are settled and fixed. > If WM wisheds to claim that there is some n for which the length of the > diagonal is NOT greater than n, let him produce it now, or forever hold > his peace. I claim that there cannot be a diagonal element where lines are lacking. That means: There are exactly as many lines as there are diagonal elements. And there are lines as long as the diagonal is. Because without lines the diagonal cannot exist. This hidden assumption leads to the necessity of aninfinite line (number) if the diagonal (the set of numbers) is infinite. This concluison excludes an actually infinite diagonal. === Subject: Re: Cantor Confusion If WM wisheds to claim that there is some n for which the length of the > diagonal is NOT greater than n, let him produce it now, or forever hold > his peace. I claim that there cannot be a diagonal element where lines are > lacking. That means: There are exactly as many lines as there are > diagonal elements. And, then we leap over a chasm to the next statement: > And there are lines as long as the diagonal is. > Because without lines the diagonal cannot exist. Without lines the diagonal cannot exist doesn't really seem to show that there are lines as long as the diagonal is. > This hidden assumption leads to the necessity of aninfinite line > (number) if the diagonal (the set of numbers) is infinite. > This concluison excludes an actually infinite diagonal. Or, it shows that the hidden assumption is contradictory. -- Marcus === Subject: Re: Cantor Confusion > I have yet to see any WM proof that is mathematically or logically > satisfactory. There are always hidden assumptions that are unwarranted. I use the hidden assumption that there are not more natural numbers > d_nn than natural numbers n. If you can accept that the are more > natural numbers than natural numbers, then we have different world > views which do not fit together and cannot be united. It is not hidden that every set that is bijectable with the naturals is bijectable with the naturals. So we should stop here. Your mathematics is not mine and vice versa. I > assume that part of my mathematics belongs to absolute truth which does > not rely on arbitrary axioms. List all of your absolute truths as if they were axioms and you will have a system which mathematicans can logically discuss. Keep them hidden, as you do, and you have nothing of any mathematical value whatsever. > Anyhow it is not useful that we comment the contributions of each other > any longer, because the positions are settled and fixed. > If WM wisheds to claim that there is some n for which the length of the > diagonal is NOT greater than n, let him produce it now, or forever hold > his peace. I claim that there cannot be a diagonal element where lines are > lacking. That means: There are exactly as many lines as there are > diagonal elements. Which is totally irrelevant to the issue of the lengths of those finite lines in comparison to the length of the infinitely long diagonal. > And there are lines as long as the diagonal is. Name one. > Because without lines the diagonal cannot exist. If there were only finitely many lines, each of length equal to its line number, then there would be one and only one line of length equal to the length of the finite diagonal. When there are infinitely many finite lines, each of length equal to its line number, then the diagonal will have infinitely many columns, one for each line, and must, therefore, be longer than every finite line. Failure to recognize and acknowledge this puts WM in Limbo, mathematically at least. === Subject: Re: Cantor Confusion > [...] >> And there are lines as long as the diagonal is. > > Name one. The names can be found in the set of finished emptiness. F. N. -- xyz Supersedes: <455ee582$0$97218$892e7fe2@authen.yellow.readfreenews.net> === Subject: Re: Cantor Confusion >> Therefore a set of ordinal number omega must exist (it is the first >> column) >> omega is not the first column. What you may write is, that in >> your sketch the first column _represents_ omega. That is a matter of taste. You are in error. Set theory under discussion does not deal with columns. > I is not uncommon to say N = omega. Since N and omega sometimes denote the same entity you may equate N with omega. There is still no column. > On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. > Omega is the first transfinite number. Omega is the first transfinite _ordinal_ number. > You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. A treatise in which variables (n) and number symbols (0, 1, ...) do not refer to sets is not a treatise _on_ set theory but a treatise of _application_ of set theory, if ever. If (!) that application fails to yield the desired result, you cannot put the blame on set theory. > This all shows that omega is a number. Set theory is not really about numbers but about sets. This all does by no means show that omega is a plain-vanilla (i.e. _natural_) number. > It is said to be the number of natural numbers. In contemporary set theory it is said that omega is _the_ _set_ of natural numbers [as Virgil pointed out the _ordered_ set]. The number (cardinality) of omega is named aleph_0. So it is said that the number (cardinality) of _the_ _set_ of natural numbers is aleph_0. The natural numbers and the set of natural numbers are two names for two different things. This linguistically reflects the difference beween An(n e omega -> p(n)) and p(omega). If you deny that convention there is hardly a common ground for debate. > And omega > n for every n e N. N = omega & a < b := a e b An(n e omega -> n < omega) I can't spot any contradiction. >> Of course not. Two sets X and Y are bijected by pairing the >> *elements* of the sets not by pairing the sets itself. If there should be infinitely many lines, then their number should be > omega. The lists and matrices are creatures of your mind. You should explain how many lines you intend to bestow upon the list. > If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d_ii) i e omega can or must be longer than every (finite) line occupancy. What every clear-thinking person agrees upon is that the cardinality of the sequence (d_ii) i e omega is greater than the cardinality of (the occupancy of) every single line of that list or matrix. This is due to the _fact_ that there are only finitely many occupied memebers in each line-sequence. > then you say that there are more natural numbers (elements > of the bijection d_nn) than natural numbers (indexes n). I don't see your conclusion. > Then there is no reason to continue this discussion. Otherwise we have > a contradiction. If clarifications are not convenient to you stop discussing whenever you want. But don't ask me not to comment your writings now and then. >> But an explanation: The set of all sets is an impossible set. >> This mathematically reads: There is no set of all sets (in some >> axiomatic systems). Shall I conclude from your definition by >> example, that you define >> possible := exists >> impossible := does not exist There is no other interpretation possible, I think. Why do you hesitate? > Sets (including bijections) are neither possible nor impossible. > They exist or do not exist. >> If they exist, then they are possible. If not, then they are >> impossible. >> Shall I interpret this as possible := exists, impossible := does not >> exist? So why don't you simply use the established terms exist and >> not exist? Maliciousness? Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no Zitierwissenschaft (quotation(s)/citation(s) science). > We should pay a bit more respect to the creator of set theory: > An mehreren Stellen meiner Arbeit werden Sie die Ansicht > ausgesprochen finden, da§ dies unm.9agliche, d. h. in sich > widersprechende Gedankendinge sind, ... I don't discuss Cantor's remarks. >> Now the part which you have not answered: > I would plead to stop this discussion. We have arrived at a clear > result. You may stop replying to my posts at any time without reason. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. > As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. ends of terms of lines is your wording not mine. > In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform you of facts. > you could simply say: There are more natural numbers d_nn than natural numbers n. Your wording not mine. > This sentence is as true as the sentence: The diagonal (d_nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have _you_ ever been arguing on a logical basis with sound arguments? ,----[ <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> ] | Adding one element (named x) to every initial segment gives us: | | ISOFTC' := { {1, x}, {1, 2, x}, {1, 2, 3, x}, ... } | | I have no clue what precisely you mean by adding one element to every | line. Do you mean the set | | L' := { 1, 2, 3, ..., x } ? | | > The fact that it is not maintained proves that your asserted | > bijection does ot exist. | | There exists a bijection between ISOTFC' and L' : | | B' := { <{1, x}, x>, <{1, 2, x}, 1>, <{1, 2, 3, x}, 2>, ... } | | proving |ISOTFC'| = |L '|. `---- F. N. -- xyz === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455db94f$0$97241$892e7fe2@authen.yellow.readfreenews.net> <455ee80f$0$97218$892e7fe2@authen.yellow.readfreenews.net> omega is not the first column. What you may write is, that in >> your sketch the first column represents omega. That is a matter of taste. You are in error. Set theory under discussion does not deal with > columns. This column contains all natural numbers in their natural order. It is N, it is omega, written horizontally. > On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. which you, unfortunately, don't know any better than its history. Omega is the first transfinite number. Omega is the first transfinite ordinal number. > Please look it up in any modern text book. You will find there that omega is a cardinal number too. > You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. In ZFC everything is a set. Not in every set theory. A treatise in which variables (n) and number symbols (0, 1, ...) > do not refer to sets is not a treatise on set theory but > a treatise of application of set theory, if ever. Don't mistake set theory with ZF or ZFC. > It is said to be the number of natural numbers. In contemporary set theory it is said that omega is the set of > natural numbers [as Virgil pointed out the ordered set]. The number > (cardinality) of omega is named aleph 0. So it is said that the number > (cardinality) of the set of natural numbers is aleph 0. You are completely in error. The number (Anzahl) of a set is its ordinal number. The cardinal number is something different (M.8achtigkeit). But even the cardinal number can be denoted by the least ordinal which can be put in bijection with a set of that number class. If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d ii) i e omega can or must be longer > than every (finite) line occupancy. What every clear-thinking person > agrees upon is that the cardinality of the sequence (d ii) i e omega is > greater than the cardinality of (the occupancy of) every single line of > that list or matrix. This is due to the fact that there are only > finitely many occupied memebers in each line-sequence. I agree. But on the other hand, the cardinality of the sequence (d nn), i. e. omega, cannot be greater than the cardinality of every single line of that list or matrix. This is due to the fact that there are only such d nn for which an n exists. If omega is a number which can be completed and even surpassed, then there must be at least one line with omega units. >> possible := exists >> impossible := does not exist There is no other interpretation possible, I think. Why do you hesitate? It is politeness. I could have said: Everybody whose intelligence is not too far below the average level will understand this interpretation. But I didn't. >> Shall I interpret this as possible := exists, impossible := does not >> exist? So why don't you simply use the established terms exist and >> not exist? Maliciousness? Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no > Zitierwissenschaft (quotation(s)/citation(s) science). Sometimes it is necessary to quote. In particular if you are uninformed but nevertheless refuse to take advice from me. In ZFC the cardinality of a set S is the least ordinal alpha such that there is a bijection from alpha to S. We should pay a bit more respect to the creator of set theory: > An mehreren Stellen meiner Arbeit werden Sie die Ansicht > ausgesprochen finden, da§ dies unm.9agliche, d. h. in sich > widersprechende Gedankendinge sind, ... I don't discuss Cantor's remarks. > Then, please stop to complain when you do not understand my expressions. I do discuss Cantor's remarks. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. It is not your wording, because you like to veil your inconsistencies, but it is your opinion. As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. ends of terms of lines is your wording not mine. It is not your wording, because you prefer to veil your inconsistencies, but it is your opinion. In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform > you of facts. Why then do you complain about the wording the diagonal of a matrix can be longer than every line? you could simply say: There are more natural numbers d nn than natural numbers n. Your wording not mine. Can you understand mathematical symbols? Can you understand d nn <--> n? This sentence is as true as the sentence: The diagonal (d nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have you ever been arguing on a logical basis with sound arguments? Yes, but that is very different from what you erroneously consider to be logical and sound. === Subject: Re: Cantor Confusion > >> omega is not the first column. What you may write is, that in >> your sketch the first column represents omega. That is a matter of taste. You are in error. Set theory under discussion does not deal with > columns. This column contains all natural numbers in their natural order. It is > N, it is omega, written horizontally. A column as a set of elements may biject with N as a set of elements, but that does not make that column and N into the same set. And columns are only horizontal when they have fallen over. On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. which you, unfortunately, don't know any better than its history. And which WM, by comparison, does not know at all. Omega is the first transfinite number. Omega is the first transfinite ordinal number. Please look it up in any modern text book. You will find there that > omega is a cardinal number too. Depends on the book. Omega has the smallest non-finite cardinality, but in some books it is not itself a cardinal. You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. In ZFC everything is a set. Not in every set theory. Which set theory does WM use in which it is otherwise? A treatise in which variables (n) and number symbols (0, 1, ...) > do not refer to sets is not a treatise on set theory but > a treatise of application of set theory, if ever. Don't mistake set theory with ZF or ZFC. Then which set theory does WM use? It is said to be the number of natural numbers. In contemporary set theory it is said that omega is the set of > natural numbers [as Virgil pointed out the ordered set]. The number > (cardinality) of omega is named aleph 0. So it is said that the number > (cardinality) of the set of natural numbers is aleph 0. You are completely in error. The number (Anzahl) of a set is its > ordinal number. The cardinal number is something different > (M.8achtigkeit). But even the cardinal number can be denoted by the > least ordinal which can be put in bijection with a set of that number > class. But need not be. And even so, there is a difference between can be denoted by and is. Such distinctions are important in mathematics, and are ignored at one's peril. > If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d ii) i e omega can or must be longer > than every (finite) line occupancy. What every clear-thinking person > agrees upon is that the cardinality of the sequence (d ii) i e omega is > greater than the cardinality of (the occupancy of) every single line of > that list or matrix. This is due to the fact that there are only > finitely many occupied memebers in each line-sequence. I agree. But on the other hand, the cardinality of the sequence (d nn), > i. e. omega, cannot be > greater than the cardinality of every single line of that list or > matrix. WM seems to be conflating the set of lines with individual lines. The number of terms in the diagonal equals the number of lines, but note that the number of lines exceeds the length of any one line, at least unless there is a last line and therefore only a finite number of lines. > If omega is a number which can be completed and even surpassed, then > there must be at least one line with omega units. False.Trivially false. Stupidly false. It is politeness. I could have said: Everybody whose intelligence is > not too far below the average level will understand this > interpretation. But I didn't. Wm certainly didn't understand any correct interpretation. > Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no > Zitierwissenschaft (quotation(s)/citation(s) science). Sometimes it is necessary to quote. In particular if you are uninformed > but nevertheless refuse to take advice from me. Those who decline taking advice from you, particularly on matters on which you are so obviously wrong, are just using their native good judgment. > In ZFC the cardinality > of a set S is the least ordinal alpha such that there is a bijection > from alpha to S. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. It is not your wording, because you like to veil your inconsistencies, > but it is your opinion. Our claim is that the diagonal of the infinite list we have been discussing is necessarily longer than any single line, because it is at least as long as the next line, and there always is a next line in an infinite list. As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. The diagonal entries are certainly determined by the end of line entries in a list in which each line is of length equal to its line index. Does WM dispute this? ends of terms of lines is your wording not mine. It is not your wording, because you prefer to veil your > inconsistencies, but it is your opinion. WRONG as usual. It is WM's claim and no one else's. In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform > you of facts. Why then do you complain about the wording the diagonal of a matrix > can be longer than every line? Every is misleading when any is intended. you could simply say: There are more natural numbers d nn than natural numbers n. > Your wording not mine. Can you understand mathematical symbols? Can you understand d nn <-- n? We can understand why you want that misinterpretation, but it is still a misinterpretation. What we mean is that there is no line as long as the diagonal. Our justification is that each line is of finite length but the diagonal is not of finite length. This sentence is as true as the sentence: The diagonal (d nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have you ever been arguing on a logical basis with sound arguments? Yes, but that is very different from what you erroneously consider to > be logical and sound. Since both mathematics and logic support our view and refute WM's, it is only in his lonely opinion that he is right and the rest of the world is wrong. === Subject: Re: Cantor Confusion > omega is not the first column. What you may write is, that in > your sketch the first column _represents_ omega. >> That is a matter of taste. >> You are in error. Set theory under discussion does not deal with >> columns. This column contains all natural numbers in their natural order. It is > N, it is omega, written horizontally. Set theory still does not deal with columns. You talk about columns. >> On the other hand omega represents what before Cantor was commonly >> abbreviated by oo. >> Irrelevant to contemporary set theory. which you, unfortunately, don't know any better than its history. Still irrelevant. >> Omega is the first transfinite number. >> Omega is the first transfinite _ordinal_ number. > Please look it up in any modern text book. You will find there that > omega is a cardinal number too. Anyway, omega is not a /natural/ number. >> You need not interpret n as a set (though you can do it). >> In contemporary set theory almost everything is a set. In ZFC everything is a set. Not in every set theory. I don't want to debate about the axioms of ZFC being sets. The point is that you want to talk about columns which neither ZFC nor any other contemporary set theory is about. >> A treatise in which variables (n) and number symbols (0, 1, >> ...) do not refer to sets is not a treatise _on_ set theory but >> a treatise of _application_ of set theory, if ever. Don't mistake set theory with ZF or ZFC. A theory of _sets_ is not a theory of _columns_. >> It is said to be the number of natural numbers. >> In contemporary set theory it is said that omega is _the_ _set_ of >> natural numbers [as Virgil pointed out the _ordered_ set]. The number >> (cardinality) of omega is named aleph_0. So it is said that the >> number (cardinality) of _the_ _set_ of natural numbers is aleph_0. You are completely in error. The number (Anzahl) of a set is its > ordinal number. Anzahl is your (or a historical person's) wording not mine. I can't spot any error in my wording. > The cardinal number is something different (M.8achtigkeit). What do you want to posit? > But even the cardinal number can be denoted by the least ordinal which > can be put in bijection with a set of that number class. Can you rephrase that? >> If you say that the diagonal can or even must be longer than >> every line, >> I never said that a diagonal (d_ii) i e omega can or must be >> longer than every (finite) line occupancy. What every clear-thinking >> person agrees upon is that the cardinality of the sequence (d_ii) i e >> omega is greater than the cardinality of (the occupancy of) every >> single line of that list or matrix. This is due to the _fact_ >> that there are only finitely many occupied memebers in each >> line-sequence. I agree. But on the other hand, the cardinality of the sequence > (d_nn), i. e. omega, cannot be greater than the cardinality of every > single line of that list or matrix. Could you rephrase (formalize) what you mean by cannot be greater? > This is due to the _fact_ that there are only such d_nn for > which an n exists. If omega is a number which can be completed and even surpassed, Define can be completed and even surpassed. > then there must be at least one line with omega units. possible := exists > impossible := does not exist >> There is no other interpretation possible, I think. >> Why do you hesitate? It is politeness. I could have said: Everybody whose intelligence is > not too far below the average level will understand this > interpretation. But I didn't. Do you understand your interpretation? > Shall I interpret this as possible := exists, impossible := does > not exist? So why don't you simply use the established terms > exist and not exist? Maliciousness? >> Knowledge of literature. Cantor. >> As you have been told quite some time before: Mathematics is no >> Zitierwissenschaft (quotation(s)/citation(s) science). Sometimes it is necessary to quote. In particular if you are > uninformed but nevertheless refuse to take advice from me. I don't need any advice from you. > In ZFC the cardinality of a set S is the least ordinal alpha such that > there is a bijection from alpha to S. What is the cardinality of 0. {} 1. { n | n < k } k e omega? 2. omega? 3. omega u { omega }? >> We should pay a bit more respect to the creator of set theory: >> An mehreren Stellen meiner Arbeit werden Sie die Ansicht >> ausgesprochen finden, da§ dies unm.9agliche, d. h. in sich >> widersprechende Gedankendinge sind, ... >> I don't discuss Cantor's remarks. > Then, please stop to complain when you do not understand my > expressions. I do discuss Cantor's remarks. Then please do not claim that ZFC is contradictory. The C in ZFC is not for Cantor. >> You claim that the diagonal of a matrix can be longer than >> every line. >> This is your wording not mine. It is not your wording, because you like to veil your inconsistencies, > but it is your opinion. My opinion is A (n e omega & |{0, 1, 2, ..., n}| < |omega|) >> As the diagonal is defined to consist of the ends of terms >> of lines, this claim is easy to conradict. >> ends of terms of lines is your wording not mine. It is not your wording, because you prefer to veil your > inconsistencies, but it is your opinion. My opinion is |(d_nn) n e omega| = |omega| >> In order to defend your claim that there are infinitely many finite >> numbers, >> In the lack of an effective offense I don't have to defend but to >> inform you of facts. Why then do you complain about the wording the diagonal of a matrix > can be longer than every line? Since even after replacing can by exists _your_ sentence does not make sense to me. There is no can in set theory. >> you could simply say: >> There are more natural numbers d_nn than natural numbers n. >> Your wording not mine. Can you understand mathematical symbols? Can you understand d_nn <-- n? Writing down a bunch of symbols does not mean that you created a mathematical notation. Rephrase what precisely There are more natural numbers d_nn than natural numbers n. shall mean. F. N. -- xyz === Subject: Re: Cantor Confusion >> Therefore a set of ordinal number omega must exist (it is the first >> column) >> omega is not the first column. What you may write is, that in >> your sketch the first column _represents_ omega. That is a matter of taste. You are in error. Set theory under discussion does not deal with columns. > I is not uncommon to say N = omega. Since N and omega sometimes denote the same entity you may equate N with omega. There is still no column. > On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. > Omega is the first transfinite number. Omega is the first transfinite _ordinal_ number. > You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. A treatise in which variables (n) and number symbols (0, 1, ...) do not refer to sets is not a treatise _on_ set theory but a treatise of _application_ of set theory, if ever. If (!) that application fails to yield the desired result, you cannot put the blame on set theory. > This all shows that omega is a number. Set theory is not really about numbers but about sets. This all does by no means show that omega is a plain-vanilla (i.e. _natural_) number. > It is said to be the number of natural numbers. In contemporary set theory it is said that omega is _the_ _set_ of natural numbers. The number (cardinality) of omega is named aleph_0. So it is said that the number (cardinality) of _the_ _set_ of natural numbers is aleph_0. The natural numbers and the set of natural numbers are two names for two different things. This linguistically reflects the difference beween An(n e omega -> p(n)) and p(omega). If you deny that convention there is hardly a common ground for debate. > And omega > n for every n e N. N = omega & a < b := a e b An(n e omega -> n < omega) I can't spot any contradiction. >> Of course not. Two sets X and Y are bijected by pairing the >> *elements* of the sets not by pairing the sets itself. If there should be infinitely many lines, then their number should be > omega. The lists and matrices are creatures of your mind. You should explain how many lines you intend to bestow upon the list. > If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d_ii) i e omega can or must be longer than every (finite) line occupancy. What every clear-thinking person agrees upon is that the cardinality of the sequence (d_ii) i e omega is greater than the cardinality of (the occupancy of) every single line of that list or matrix. This is due to the _fact_ that there are only finitely many occupied memebers in each line-sequence. > then you say that there are more natural numbers (elements > of the bijection d_nn) than natural numbers (indexes n). I don't see your conclusion. > Then there is no reason to continue this discussion. Otherwise we have > a contradiction. If clarifications are not convenient to you stop discussing whenever you want. But don't ask me not to comment your writings now and then. >> But an explanation: The set of all sets is an impossible set. >> This mathematically reads: There is no set of all sets (in some >> axiomatic systems). Shall I conclude from your definition by >> example, that you define >> possible := exists >> impossible := does not exist There is no other interpretation possible, I think. Why do you hesitate? > Sets (including bijections) are neither possible nor impossible. > They exist or do not exist. >> If they exist, then they are possible. If not, then they are >> impossible. >> Shall I interpret this as possible := exists, impossible := does not >> exist? So why don't you simply use the established terms exist and >> not exist? Maliciousness? Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no Zitierwissenschaft (quotation(s)/citation(s) science). > We should pay a bit more respect to the creator of set theory: > An mehreren Stellen meiner Arbeit werden Sie die Ansicht > ausgesprochen finden, da§ dies unm.9agliche, d. h. in sich > widersprechende Gedankendinge sind, ... I don't discuss Cantor's remarks. >> Now the part which you have not answered: > I would plead to stop this discussion. We have arrived at a clear > result. You may stop replying to my posts at any time without reason. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. > As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. ends of terms of lines is your wording not mine. > In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform you of facts. > you could simply say: There are more natural numbers d_nn than natural numbers n. Your wording not mine. > This sentence is as true as the sentence: The diagonal (d_nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have _you_ ever been arguing on a logical basis with sound arguments? ,----[ <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> ] | Adding one element (named x) to every initial segment gives us: | | ISOFTC' := { {1, x}, {1, 2, x}, {1, 2, 3, x}, ... } | | I have no clue what precisely you mean by adding one element to every | line. Do you mean the set | | L' := { 1, 2, 3, ..., x } ? | | > The fact that it is not maintained proves that your asserted | > bijection does ot exist. | | There exists a bijection between ISOTFC' and L' : | | B' := { <{1, x}, x>, <{1, 2, x}, 1>, <{1, 2, 3, x}, 2>, ... } | | proving |ISOTFC'| = |L '|. `---- F. N. -- xyz === Subject: Mathsoft's licensing, grr.. Hello all, I changed a drive today and now Mathcad 13 wants me to reactivate. Problem is that mathcad.com does not resolve and I can't do my work. I tried from my PC and a university computer in a different state. And of course http://www.mathsoft.com doesn't work either. - Robert === Subject: What Do Muslims Believe about Jesus? Muslims respect and revere Jesus (peace be upon him). They consider him one of the greatest of God's messengers to mankind. The Quran confirms his virgin birth, and a chapter of the Quran is entitled 'Maryam' (Mary). The Quran describes the birth of Jesus as follows: (Remember) when the angels said, O Mary, God gives you good news of a word from Him (God), whose name is the Messiah Jesus, son of Mary, revered in this world and the Hereafter, and one of those brought near (to God). He will speak to the people from his cradle and as a man, and he is of the righteous. She said, My Lord, how can I have a child when no mortal has touched me? He said, So (it will be). God creates what He wills. If He decrees a thing, He says to it only, 'Be!' and it is. (Quran, 3:45-47) Jesus was born miraculously by the command of God, the same command that had brought Adam into being with neither a father nor a mother. God has said: The case of Jesus with God is like the case of Adam. He created him from dust, and then He said to him, Be! and he came into being. (Quran, 3:59) During his prophetic mission, Jesus performed many miracles. God tells us that Jesus said: I have come to you with a sign from your Lord. I make for you the shape of a bird out of clay, I breathe into it, and it becomes a bird by God's permission. I heal the blind from birth and the leper. And I bring the dead to life by God's permission. And I tell you what you eat and what you store in your houses.... (Quran, 3:49) Muslims believe that Jesus was not crucified. It was the plan of Jesus' enemies to crucify him, but God saved him and raised him up to Him. And the likeness of Jesus was put over another man. Jesus' enemies took this man and crucified him, thinking that he was Jesus. God has said: ...They said, We killed the Messiah Jesus, son of Mary, the messenger of God. They did not kill him, nor did they crucify him, but the likeness of him was put on another man (and they killed that man)... (Quran, 4:157) Neither Muhammad nor Jesus came to change the basic doctrine of the belief in one God, brought by earlier prophets, but rather to confirm and renew it.1 === Subject: Re: What Do Muslims Believe about Jesus? > Muslims respect and revere Jesus (peace be upon him). They consider > him one of the greatest of God's messengers to mankind. The Quran > confirms his virgin birth, and a chapter of the Quran is entitled > 'Maryam' (Mary). The Quran describes the birth of Jesus as > follows: > (Remember) when the angels said, O Mary, God gives you good news of > a word from Him (God), whose name is the Messiah Jesus, son of Mary, > revered in this world and the Hereafter, and one of those brought near > (to God). He will speak to the people from his cradle and as a man, > and he is of the righteous. She said, My Lord, how can I have a > child when no mortal has touched me? He said, So (it will be). > God creates what He wills. If He decrees a thing, He says to it only, > 'Be!' and it is. (Quran, 3:45-47) Jesus was born miraculously by the command of God, the same command > that had brought Adam into being with neither a father nor a mother. > God has said: The case of Jesus with God is like the case of Adam. He created him > from dust, and then He said to him, Be! and he came into being. > (Quran, 3:59) During his prophetic mission, Jesus performed many miracles. God tells > us that Jesus said: I have come to you with a sign from your Lord. I make for you the > shape of a bird out of clay, I breathe into it, and it becomes a bird > by God's permission. I heal the blind from birth and the leper. And > I bring the dead to life by God's permission. And I tell you what > you eat and what you store in your houses.... (Quran, 3:49) Muslims believe that Jesus was not crucified. It was the plan of > Jesus' enemies to crucify him, but God saved him and raised him up to > Him. And the likeness of Jesus was put over another man. Jesus' > enemies took this man and crucified him, thinking that he was Jesus. > God has said: ...They said, We killed the Messiah Jesus, son of Mary, the > messenger of God. They did not kill him, nor did they crucify him, > but the likeness of him was put on another man (and they killed that > man)... (Quran, 4:157) Heresy. Neither Muhammad nor Jesus came to change the basic doctrine of the > belief in one God, brought by earlier prophets, but rather to confirm > and renew it.1 === Subject: problem in number theory show that x^4-y^4=2z^2 has no integer solutions. Help! === Subject: Re: problem in number theory Ã/¸ champagne 101@yahoo.com Ûçòáãå: > show that x^4-y^4=2z^2 has no integer solutions. > For x=0,y=0,z=0 it is true that x^4-y^4=2z^2 and since 0 is an integer what you say can't be proven.... === Subject: Re: problem in number theory > show that x^4-y^4=2z^2 has no integer solutions. Help! There exist solutions with z=0 or y=0. To prove that no other solutions exist, it suffices to prove there are no positive integer solutions. It may be assumed that x and y are coprime. Therefore they are both odd. Then (x^2-y^2)(x^2+y^2)=2z^2 x^2-y^2=s^2, x^2+y^2=2t^2 (x-y)/2=u^2, (x+y)/2=v^2. Then we get t^2=u^4+v^4. But this equation was shown to be impossible by Fermat. === Subject: Re: JSH: Connecting the dots > You people need your own lives versus getting so worked up about mine. Does laughing hysterically count as getting worked up? === Subject: Re: Math as Religion > >In this case, the word is redundant with prove. Feel free to delete >it. Well the problem is not whether I should feel free to delete it but > whether you should feel free to use it. Obviously you do. So it seems > you would argue that prove and true are coincident, In mathematics. > meaning you > regard your axiomatic mathematical assumptions of truth as true. No idea how you jumped to that erroneous conclusion. If by mathematical assumptions of truth, you mean axioms, where did I ever say that we prove axioms? -- Marcus === Subject: Re: Math as Religion >> And perhaps you'd be so good as to define correct and incorrect >> mathematics for the heathen? Correct means the definitions and theorems are clearly stated and the >proofs are all valid with each step following from previous steps or >axioms. Incorrect means not correct. Well I'll buy the last. But the explanation for correct is defective > if we have all these things and the result is still incorrect. Nothing > in your definition for correct mathematics demonstrates the initial > assumptions are true only that they follow from whatever assumptions > one makes. Yes. Right. That is what correct mathematics is. -- Marcus === Subject: Re: Math as Religion A Google search for Penrose Godel will turn up lots of interesting > reading. Perhaps. > It will not however turn up the obvious error in The Emperor's New Mind > (at least you seem to have discovered what book you are talking about - > previously the error was just in a book by Penrose). > As this error is - according to you - easy to spot, > it should not find you long to locate it, and share it with us. I already told you. Penrose concludes from the proof of Godel's Theorem that we know the Godel sentence is true, even though ZFC can't prove it. (algorithms/logical systems) can't do. -- Marcus === Subject: Re: Math as Religion If you don't wish to believe Penrose is in error, that's your > prerogative. It isn't a matter of belief. > You have not told us what this obvious error is. > If and when you do, I shall form an opinion on it. Penrose's argument goes basically like this: 1. Mathematicians can do X. 2. Mathematicians can show that machines/algorithms cannot do X. 3. Mathematicians are people. 3. Therefore people can do things that machines can't do. I believe you said that you read his book and found it to be correct. If so, please tell us what X is. > Plus, Penrose is a crank, so non-cranks could reasonably be expected to > be reluctant to appear on a panel with him. It would be like inviting a > proponent of intelligent design to be on a panel on the future of > biology. I don't believe Penrose (whom I know slightly) > is even mildly crankish. Define crankish. > He has made major contributions in a number of fields - > the generalized inverse of a matrix, non-periodic tiling, > as well of course as general relativity. I never said otherwise. > He has annoyed proponents of AI, certainly, > but that does not make him a crank. Of course not, if that was all he did. > In fact I would say that his negative views on AI are now shared > by a large majority of mathematicians. Do you have any evidence for this belief? I'm not aware of anything written in a math journal that would support this, and I have quite a few things written in math journals that disagree with his views. -- Marcus === Subject: Re: We, as Scientists must show tolerance to opposite views > Another great example of your psychosis... And that's supposedly the all-knowing and/or outter-limits as to your buttology spewing infomercial expertise? My somewhat dyslexic encrypted interest in this topic is in seeing that most anything using the regular laws of physics and of whatever can be reasonably replicated should be allowed and shared, as best can be accomplished, whereas you and others of your warm and fuzzy kind obviously think otherwise. - Brad Guth -- === Subject: Re: MathCAD problem need help please > Hi people, How I can sum up an array? > Following is my code x:=0,.5..2 > y(x):=x+1 y(x)= > 1 > 1.5 > 2 > 2.5 > 3 Now I want to sum up the y(x) (1+1.5+2+2.5+3) > How I can do that? x:=2..6 Summation x/2