Cantor's Diagonal?
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From: Virgil on 10 Feb 2010 02:29 In article <cce72132-0a48-4a81-9ff2-7a352a45d758(a)u5g2000prd.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > On Feb 9, 10:34�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <a8554fc6-0b17-44e5-88dc-4543a9338...(a)s33g2000prm.googlegroups.com>, > > �"Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > The equivalency function, though, that works in general models of the > > > reals, and variously as I show in the proof machinery of results about > > > uncountability of the reals not similarly to any other function you > > > describe, by proof, still sees that you are selecting a different by > > > the symmetry of the diagonal, with respect to EF's about the other. > > > > �Would you translate that into comprehensible English please. > > The symmetry about the diagonal has the notion that there is the > opposite diagonal in the matrix. In the so called "diagonal" construction there is no symmetry implied or any actual symmetry required. > Now, there is no end of the rows and > columns of the infinite matrix with a corner. There is no need of any infinite matrix at all. All that is needed is an enumerated enumerable set of binary functions on N. > Yet, there is a natural > representation in finite aproximation. > > Then, where Cauchy sequences underdefine reals If anything, Cauchy sequences would considerably overdefine the reals as there are at least countably many such sequences defining each real. What you need is Cauchy sequences of rationals "Mod" the null sequences, i.e., equivalence classes of Cauchy sequences modulo the equivalence relation in which two seqeunces are equivalent if a and only if their difference converges to 0. The remainder is too vague even to be correctable, so I snipped it.
From: Ross A. Finlayson on 10 Feb 2010 03:46 On Feb 9, 11:29 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <cce72132-0a48-4a81-9ff2-7a352a45d...(a)u5g2000prd.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > On Feb 9, 10:34 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <a8554fc6-0b17-44e5-88dc-4543a9338...(a)s33g2000prm.googlegroups.com>, > > > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > The equivalency function, though, that works in general models of the > > > > reals, and variously as I show in the proof machinery of results about > > > > uncountability of the reals not similarly to any other function you > > > > describe, by proof, still sees that you are selecting a different by > > > > the symmetry of the diagonal, with respect to EF's about the other. > > > > Would you translate that into comprehensible English please. > > > The symmetry about the diagonal has the notion that there is the > > opposite diagonal in the matrix. > > In the so called "diagonal" construction there is no symmetry implied or > any actual symmetry required. > No, that's false, mathematically, because it's implied, because as there is comparison of progression of the diagonal, for the other diagonal in this case, where the forward diagonal is properly a main diagonal, the numeric rules about difference in expansion for each, expansion as matrix rows, in terms of rules defined by numbering operations, they're about Cantor's diagonal or the anti-diagonal, or an anti-diagonal, to see induction over the elements of the list, and as well exhaustion of induction over the elements of the list, it's a forward rule. The other diagonal and what it represents about symmetry exists and is perpendicular to the matrix for each element of the diagonal. Also it is as I generally describe, in the equivalency function, where otherwise generally the reals are uncountable, in terms of representing the real numbers that are elements of the range of this function, as modeled standardly by real functions, if they had expansions: all the elements off of the finite induction rule would only have zeroes in their region against which to exhaust the space in terms of constructing anything other the antidiagonal. > > Now, there is no end of the rows and > > columns of the infinite matrix with a corner. > > There is no need of any infinite matrix at all. All that is needed is an > enumerated enumerable set of binary functions on N. > > > Yet, there is a natural > > representation in finite aproximation. > > > Then, where Cauchy sequences underdefine reals > > If anything, Cauchy sequences would considerably overdefine the reals as > there are at least countably many such sequences defining each real. > Yeah, and uncountably many for each. To say they underdefine, there are multiple definitions of define in mathematics. When I say it here, it means that in terms of the complete space of those things, whether they represent all of the real numbers given their normal properties defined by Cauchy sequences. > What you need is Cauchy sequences of rationals "Mod" the null sequences, > i.e., equivalence classes of Cauchy sequences modulo the equivalence > relation in which two seqeunces are equivalent if a and only if their > difference converges to 0. > Those are constructible. Here, the null sequences are just the elements of the equivalence class. They're constructible means that the Cauchy sequences of rationals are constructible. Also they're countable. > The remainder is too vague even to be correctable, so I snipped it. Here again, another thread where the equivalency function is considered and it is shown how in modern, standard mathematical parlance, it applies. Thanks, Ross Finlayson
From: Herman Jurjus on 10 Feb 2010 04:30 Transfer Principle wrote: > > The main point here is not to attack ultrafinitism, but merely > to point out that if I hold Y-V and WM to my same standards, > then either both are "cranks," or neither are. So much for your standards, then. The problem with WM is not so much that he's 'wrong', but that he's a lousy mathematician. He's confused about lots of things: He confuses rejection of actual infinity with ultra-finitism. He claims to have proved ZFC inconsistent with a proof that's not formalizable in ZFC (and doesn't show any signs of being aware of the latter). He claims he agrees with Robinson where he doesn't. Etc, etc. But worst of all: he doesn't even attempt a formalization or precise elucidation of what is and is not allowed in his mathematics. A rather big contrast with Yessenin-Volpin, wouldn't you say? -- Cheers, Herman Jurjus
From: Herman Jurjus on 10 Feb 2010 06:34 Transfer Principle wrote: > On the other hand, the following WM quote appears in the > thread to which I linked above: > >> "Axiom Of Potential Infinity: For every natural number there is a set >> that contains this number together with all smaller natural numbers, >> and for every set of natural numbers there is a natural number that >> is larger than every number of the set." > > In this post, WM has expressed that he wants potentially > infinite sets to exist, but for no _actually_ infinite > sets to exist. It matters little to "cranks" like WM > that without actually infinite sets, calculus is most > likely impossible. On the contrary. For all practical purposes, all of it can still be done, be it in a more clumsy way. (After all, before 1908, all mathematics had to be done without relying on infinite sets.) -- Cheers, Herman Jurjus
From: Carsten Schultz on 10 Feb 2010 07:11
Am 10.02.10 10:30, schrieb Herman Jurjus: > Transfer Principle wrote: >> >> The main point here is not to attack ultrafinitism, but merely >> to point out that if I hold Y-V and WM to my same standards, >> then either both are "cranks," or neither are. > > So much for your standards, then. > > The problem with WM is not so much that he's 'wrong', but that he's a ^^ ^^^^ > lousy mathematician. ^^^^^^^^^^^^^^ I have marked a mistake that you made, but apart from that I totally agree with you. > He's confused about lots of things: > He confuses rejection of actual infinity with ultra-finitism. > He claims to have proved ZFC inconsistent with a proof that's not > formalizable in ZFC (and doesn't show any signs of being aware of the > latter). > He claims he agrees with Robinson where he doesn't. > Etc, etc. > > But worst of all: he doesn't even attempt a formalization or precise > elucidation of what is and is not allowed in his mathematics. > > A rather big contrast with Yessenin-Volpin, wouldn't you say? -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page. |