Cantor Confusion
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From: Lester Zick on 10 Nov 2006 17:03 On 10 Nov 2006 11:05:20 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Never! I gave a mathematical proof. That has nothing to with set thery. > >David Marcus listed your exact remarks. It was clear that you >represented to give a proof in set theory. > >As I said, "I am open to seeing your argument as SET THEORETIC if you >would just cooperate". [all caps added]. But now you you say your >argument "has NOTHING to do with set theory" [all caps added]. I'm >DEFINITELY not interested in your informal arguments that have no >axioms, no primitives, no formal definitions, and no specified logic >that have NOTHING to do with set theory.. > >So let's be definite here. Do you have a strong belief that your >argument can be expressed in the language of set theory and uses only >first order logic applied to the axioms of set theory? If so, then, >your patience allowing, we can work together as you explain your >concepts and terminology to me and I strive to see how to formalize it >as a set theoretic proof. But if you really do NOT have a strong belief >that your argument can be expressed in the language of set theory and >uses only first order logic applied to the axioms of set theory, then >I'm not interested in wasting my time. > >You can let me know which it is. Your way or the highway huh, Moe(x). ~v~~
From: Virgil on 10 Nov 2006 17:04 In article <1163155686.834140.90040(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > My question is : Do you maintain omega > n for all n e N? I know that > > > modern set theory says so. If something can be larger than a number, > > > then it must be a number. But "larger" in this context merely means "contains as a member", and there are sets which are not numbers which contain numbers as members. So WM's claim that every set which contains a number is a number, is false. > > > > Omega is not an element of N. However you can compare omega with > > any element of n. > > Therefore we can compare the diagonal with every line. We find that the > diagonal is longer than every line. In the sense that the ordinal "counting" the number of columns in any finite row is a member of the ordinal "counting" all infinitely many columns. > This is about the same as the vase > which is empty at noon or Tristram Shandy who completes his diary. About both of which WM was also wrong. > > No reason to bother about this. Right! Just get used to being told you are wrong, when you are wrong. It is really no bother at all to tell you. > > Regards, WM
From: Virgil on 10 Nov 2006 17:07 In article <1163156724.726714.220060(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > > > > > Wrong.. You have not understood the meaning. We have: The set of all > > > > > sets does not exist. But if all mathematical entities including all > > > > > sets do exist in a Platonist universe, how can it be that the set of > > > > > all sets does not exist? > > > > > > > > That is NOT Fraenkel, Bar-Hillel, and Levy's point at all! > > > > > > You admitted that you do not understand their philosophical remarks. So > > > let it be. > > > > No, please do not misparaphrase me. I said that I admit that I don't > > fully MASTER their philosophical remarks. But I said that I do know > > basically what they're talking about and what they're saying. > > You may think so, but that is obviously false (see below). Since WM has so obviously poor a grasp on any of this, he is hardly in any position to criticize anyone else's grasp. >
From: Virgil on 10 Nov 2006 17:17 In article <1163157029.650628.181870(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Daniel Grubb schrieb: > > > >> Since the length of the diagonal equals the length of the first column, > > >> you are also saying that there is some line whose length is greater than > > >> or equal to the length of the first column. Is that correct? > > > > >The diagonal of actually infinite length omega cannot exist without the > > >existence of a line of actually infinite length omega. > > > > Why not? Do you have a proof of this claim? > > The diagonal of a matrix is defined as consisting of elements of this > matrix. For a diagonal longer than every line (or every column) this > is impossible. As it is only WM who insists on a "matrix" analogy (and even then infinite matrices are quite possible), it is only his problem to resolve such apparent problems. What we see is a function from N to the set of finite strings over some set of characters, with f(n) being a string of length n, with f_m(n), 1 <= m <= n being its nth character, and a "diagonal" being a function d from N to the same character set with d(n) being determined from f_n(n). Any "arrangement" into rows ( or lines) and columns is totally unnecessary.
From: David Marcus on 10 Nov 2006 17:22
Franziska Neugebauer wrote: > David Marcus wrote: > > > [...] you are working with an infinite triangle [...] > > ,----[ http://en.wikipedia.org/wiki/Triangle ] > | A triangle is one of the basic shapes of geometry: a polygon with > | three vertices [...] > `---- > > Are there really three vertices in WM's "triangle"? No. But, I don't think an "infinite triangle" needs to be a triangle. However, I'm open to suggestions for what to call it. -- David Marcus |