Cantor Confusion
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From: Virgil on 12 Nov 2006 01:05 In article <1163282191.379752.4730(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > 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? > > I need no belief. But you have all sorts of beliefs for which your only justification is other beliefs. > If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. That presumes, contrary to present evidence, that your argument is consistent with all consistent mathematics.
From: Virgil on 12 Nov 2006 01:10 In article <1163282642.872607.13390(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > You carried out a survey? Or is that your guess? I know some > > > mathematicians who know what a mathematician should understand by > > > "limit", "number", "grow". > > > > Please post their definitions of the words. > > Try to get a clue of Cantor's second creation principle. Then you will > know these things. We already know what current mathematics says about "limit" and "number", What we do not know, and cannot find out from anybody but WM, what WM means by them. > > > > > Which mathematics do you allude to? > > > > That which is taught in school, explained in textbooks, published in > > journals, and discussed by some in this newsgroup. > > I teach at a school, I wrote textbooks (and my new book is in print), I > published in journals and I wrote in newsgroups. What you teach in school, write in textbooks and publish in journals, is not, by any evidence produced here, liable to be relevant to mathematics.
From: Virgil on 12 Nov 2006 01:15 In article <1163282761.745992.253640(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > They were discussing mathematics. Philosophy belongs to mathematics. > > > > There may be some benighted mathematicians who claim so, but there are > > as many philosophers, and others, who will claim that mathematics > > belongs to philosophy. > > Did I say that *all* philosophy belongs to mathematics? The philosophy > of mathematics belongs to mathematics. > > Regards, WM A statement like "Philosophy belongs to mathematics" does not seem to leave much room for any part of philosophy to belong anywhere else. Besides which, there are philosophers, and busybodies like WM, who seem to believe that mathematics is too important to be left to mathematicians.
From: Virgil on 12 Nov 2006 01:30 In article <1163282892.731472.33380(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > William Hughes wrote: > > > > > a: Any infinite set of numbers must contain an infinite number > > > > > > b: It is possible to have an infinite set of numbers that does > > > not contain an infinite number. > > > > WM, please tell us if you agree or disagree with the statements a and b > > above. > > > You will not understand it. But I'll try. > > The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. Sets cannot be merely potential. If not actual, then not sets. Whatever it is that WM is talking about when he misspeaks of "potential sets", it is not sets. > > An actually infinite set is a set which has a cardinal number like > omega. In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. There is nothing in mathematics that says that the "number" of elements in a set must be a member of that set. WM wants to have the number of members of a natural to be a member of itself, which does not happen in ZF or NBG. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. > > The contradiction becomes obvious in the matrix discussed or in the > following few lines: > > Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a > sequence with ordinal number omega + n. > Here omega does not appear in the sequence. omega is nothing but the > sequence > n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number > alike the finite ordinals. Therefore, omega must appear in the > sequence: > n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n > What is the ordinal number of this sequence? Union( omega, { omega+1, omega+2, omega+3, ... omega+n+1})
From: Virgil on 12 Nov 2006 01:33
In article <1163283361.153576.304690(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > > > WM does not belong to the "factinista". He is ignorant about the fact > > that nowadays mathematics is a formal science in the first place. > > The formalists are like a watchmaker who is so absorbed in making his > watches look pretty that he has forgotten their purpose of telling the > time, and has therefore omitted to insert any works. (Bertrand Russell) While WM is so concerned that the works work his way that he has forgot how to tell time. |