Cantor Confusion
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From: mueckenh on 29 Nov 2006 11:17 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > That is not the question any longer. The question is: Can a set > > > > theorist admit that she is in error? It seems impossible. They all are > > > > too well trained in defending ZFC. > > > > > > In error as to what? Set theorists admit mistakes. It is not uncommon > > > for books to have errata sheets attached. > > > > Would you see a contradiction in these two statements? > > 1) "The cardinality of omega is |omega| not omega." > > 2) "The cardinality of omega, also written as |omega|, is omega". > > If you give me the context of those remarks, I might have something to > say about them. There are two possibilities. a) The two statements have been uttered by a set theorists convinced of set heory. So there is no contradiction. b) The two statements have been uttered by a critically minded Contra. Then they are rubbish. Rgards, WM
From: mueckenh on 29 Nov 2006 11:20 Randy Poe schrieb: > MoeBlee wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > That is not the question any longer. The question is: Can a set > > > > > theorist admit that she is in error? It seems impossible. They all are > > > > > too well trained in defending ZFC. > > > > > > > > In error as to what? Set theorists admit mistakes. It is not uncommon > > > > for books to have errata sheets attached. > > > > > > Would you see a contradiction in these two statements? > > > 1) "The cardinality of omega is |omega| not omega." > > > 2) "The cardinality of omega, also written as |omega|, is omega". > > > > If you give me the context of those remarks, I might have something to > > say about them. > > In its original context, statement #1 was about the NOTATION for > "cardinality of omega". No. The first statement was from a set theorist who was not aware that it is usual to denote the cardinal number aleph_0 by omega. > > In its original context, statement #2 was about a theorem > that |omega| = omega. > No. The second statement was from a set theorist who meanwhle had learnt that it is usual to denote the cardinal number aleph_0 by omega. Regards, WM
From: mueckenh on 29 Nov 2006 11:24 William Hughes schrieb: > > > The two statements: > > > > > > The elements of N satsify property X > > > and > > > The set N satisfies property X > > > > > > are independent.. Induction proves that the elements of N are finite. > > > Induction does not show that N is finite. > > > > Of course it does not. Induction is a sensible method. By induction > > only sensible assertions can be proved. The *set generated by > > induction* is always finite. > > > > Using your interpretation of "set generated by induction" > > all sets of natural numbers are finite. > if all sets of natural numbers can be generated by induction, Do you deny this assertion? > > So all we have to do to prove that all sets of natural numbers > are finite is to assume that all sets of natural numbers > can be generated by induction. > > No set of natural numbers without a largest element can > be generated by induction. Do you deny this assertion? > > So what we have to do to prove that all sets of natural numbers > are finite is to assume that there is no set of natural numbers > without a largest element. No. We have to accept thart there are sets which are capable of growing, as Fraenkel et al. expess it. Then we have finite sets without a largest element. Then we describe reality correctly. Regards, WM
From: mueckenh on 29 Nov 2006 11:29 MoeBlee schrieb: > > > > That's why Skolem liked ZFC soo much? > > > > > > Yes, Skolem was critical of set theory. > > > > What do you think, why? > > He gives reasons in his paper. > > By the way, Skolem's remarks pertain to Zermelo's set theory, and are > not addressed to ZFC itself, since that came later, and the axiom > schema of replacement in ZF remedies one of Skolem's criticisms *one* of them! > as well > as, if I'm not mistaken, by the time we get to ZF, Skolem's own > recommendation for using formulas instead of 'definite properties' had > been incorporated. > > > > > > > Why do you think that our > > > > arithmetic (which is not a model of ZFC) does contain such a bijection? > > > > > > I don't know what bijection you are referring to. Exactly what > > > bijection in exactly what set do you claim existence? > > > > No bijection in any infinite set. But why do you think that there is a > > bijection N <--> Q? > > No bijection IN any infinite set? In set theory, 'in' usually means > 'member of'. Of course there are bijections that are members of > infinite sets. > > > But why do you think that there is a > > bijection N <--> Q? > > The reason I believe it's a theorem of Z set theory that there is a > bijection between the set of natural numbers and the set of ordered > pairs of natural numbers is because I've studied at least one proof. It is believed to be a proof. But it isn't a proof. It is demonstarted for few symbols. Regards, WM
From: mueckenh on 29 Nov 2006 11:34
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > > > > If both, W and N, are countable, then renaming the elements of one of > > > > > > them leads to an identity map. > > > > > > > > > > You are confusing bijections within the model to bijections > > > > > outside of the model. > > > > > > > > And who told you that you were outside? > > > > And who told you that you were outside? > > > > > You are noting that it is not possible to construct a bijection > > > in a nonstandard model and that it is > > > possible to construct a bijection in the standard model. This > > > is true, however, it is not a contradiction. > > > > In particular because there is neither a stadard model nor a > > non-standard model of ZFC. > > You haven't proven that there is no model of ZFC. Mathematics exists nowhere except in the minds of mathematicians. At present there is no mind with such a model. So, where should it exists? > > > And in particular because the expression "contradiction" is not pat of > > ZFC. > > In Z set theory, we can formulate definitions of 'S is a contradiction' > and 'T is inconsistent'. One could, but one wouldn't. Never! Regards, WM |