Cantor Confusion
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From: mueckenh on 21 Nov 2006 16:11 Randy Poe schrieb: > > > > You never heard of countable uncountable models? > > I've never heard of something which is both countable > (can be bijected to N) and uncountable (can not be bijected > to N). That would seem to require both P and (not P) to > be true for some proposition P. Indeed, that is my impression too. But ZFC cannot tolerate P and notP. Therefore people have dispelled it. If ZFC is correct, then, according to a theorem of Skolem, it must have a countable model. But the most important theorem of ZFC proves the existence of uncountable sets. Therefore the worshippers of transfinity proclaim the existence of a set which is countable (as Skolem requires) "from outside" but does not contain the bijection with N internally. Therefore "inside the model", countability cannot be proved - and all is fine. > > > What about all contructible numbers or all words? They cannot be mapped > > on N though they are a countable set. > > The set of all finite words, the set of polynomials, or any > countable set can be put in bijection with N. Where do you > get your view that it "can't be mapped to N"? The list of all finite words cannot be constructed. The list of all constructible numbers cannot be constructed. That means, these bijections cannot be constructed. Regards, WM
From: Franziska Neugebauer on 21 Nov 2006 16:13 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: [...] >> > Therefore, the sentence "The cardinality of omega is |omega| not >> > omega" is false? Or is it not false? >> >> Perhaps D. Marcus' understanding in >> <MPG.1fcbb0f0d873ccdd989977(a)news.rcn.com> >> may help you to cope with my sentence under discussion. >> > You wrote: > 1) "The cardinality of omega is |omega| not omega." > And after learning from me that this is wrong, > 2) "The cardinality of omega, also written as |omega|, is omega". > > Now I am interested whether or not this is a contradiction in your > eyes. If you say, no, it is slightly misleading but it is not a > contradiction, then I am absolutely clear that ZFC will remain free of > contradictions forever. But then I can tell the young students who not > yet worship ZFC one more example of the logic of the worshepherds of > transfinity. Nice try. F. N. -- xyz
From: mueckenh on 21 Nov 2006 16:13 William Hughes schrieb: > The problem is the difference between saying > > The set of all natural numbers exists > and > We can generate an arbitrarily large set > of natural numbers > > (i.e. the difference between actual and potential infinity) is > mostly one of terminology. No. The first can be disproved, the second cannot be disproved. > > About the only thing you can say about the > "potentially infinite set of natural numbers" is that it is > not a set. So what? You still have this potential, > this thing that allows you to get new natural numbers > whenever you want. To produce or construct them, not to take them from the shelf. Regards, WM
From: Virgil on 21 Nov 2006 16:16 In article <1164106494.391083.305500(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Every element except the first is "outside" at least one line. > > > > While it is true that there is no diagonal element that is "outside" of > > EVERY line, that is quite a different issue. > > > > > No, just this is the decisive issue. Only to those with quantifier dyslexia. > > > > In ZF, statements with counterexamples are not theorems. > > One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line containing every element of the diagonal, which there is not. WM conflates "every element of the diagonal is in SOME line", which is true, with "every element in the diagonal is in THE SAME line", which is false.
From: Virgil on 21 Nov 2006 16:19
In article <45630207.3030502(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/9/2006 10:32 PM, Virgil wrote: > > > > > WM again uses "number" where we use "set". > > > > We have a SET which contains all of the /finite/ naturals or all of the > > /finite/ ordinals. > > Yes, modern set-mathematicians have their Gruenderzeit belief. Modern anti-set non-mathematicians claim to know what set theory should be and what mathematics should be without knowing damn all about either. |