Cantor Confusion
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From: David Marcus on 11 Nov 2006 17:35 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Since Cantor quite a bit has changed. Limits are *not* part of set > > > > theory, they belong to topology and other things build on set theory. > > > > > > Experience has shown that practically all notions used in contemporary > > > mathematics can be defined, and their mathematical properties derived, > > > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > > > include limits? > > > > Darn. Do read. It does include limits, but not in the branch of set > > theory. > > All "modern mathematics" is set theory. Set theory is not a branch but > the foundation, in fact, it is all. The fact that set theory is the foundation does not mean it is all. That isn't what the words "foundation" and "all" mean in English. -- David Marcus
From: David Marcus on 11 Nov 2006 17:36 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) I believe you are confusing the terms "formal science" and "formalists". I don't think Franziska meant them to be the same. They don't mean the same thing to me. -- David Marcus
From: William Hughes on 11 Nov 2006 18:52 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. Not a very clear attempt. Try again, starting your answer with something like "a is true and b is false" or "b is true and a is false or "it depends on which assumptions you make". > > The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. > > An actually infinite set is a set which has a cardinal number like > omega. You are confusing cardinals and ordinals. For finite sets they are the same. For infinite sets they are different. You want ordinals. This cardinal (actually ordinal) number is not an element of the set > 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. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. BZZZ 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. For example: For every set B of the form {1,2,3,...,n} there exists a finite natural number m such that B={1,2,3,...,m}. However there is no finite natural number m such that N={1,2,3,...,m} [This is true even if we assume that N consists of exaclty those natural numbers that will be named during the lifetime of the universe]. > > 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? > omega + n + 1 This is the same as the ordinal number of 1,2,3,...,omega,omega+1,omega+2,...,omega+n You assume that there is only one representation of an ordinal and get an immediate contradiction because there is in fact more than one representation of an ordinal. - William Hughes
From: David Marcus on 11 Nov 2006 19:45 William Hughes wrote: > 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. > > Not a very clear attempt. It was clearer than most of WM's attempts. Be encouraging! > Try again, starting your answer with something like "a is true and b > is false" > or "b is true and a is false or "it depends on which assumptions > you make". -- David Marcus
From: Dik T. Winter on 11 Nov 2006 19:47
In article <MPG.1fbfe58c2ccf99539898d1(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > stephen(a)nomail.com wrote: > > I think a lot of this "opposition" would go away if the word > > "transfinite" instead of "infinite" had been used to describe > > a set that can be put into a one-to-one correspondence with > > a proper subset of itself. The word "infinite" sends people > > down strange philosophical paths, as does the word "infinity" > > despite the fact that it is not really even used in set theory. > > Noone would argue about "transfinity". > > You could be right. Although, it seems unfair of the cranks to dictate > what words mathematicians can appropriate. It is hard to make up good > names. We have enough names like "second category" as it is. I think stephen is right. But as you say, there are so many other words that mathematicians do use to which there is opposition. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |