Cantor Confusion
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From: MoeBlee on 22 Nov 2006 13:22 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. MoeBlee
From: MoeBlee on 22 Nov 2006 13:31 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > If ZFC is correct, then, according > > > to a theorem of Skolem, it must have a countable model. > > > > If ZFC is consistent, then it has a countable model. > > > > If one finds that to be an unappealing feature of ZFC, then fine, one > > can choose not to use ZFC or to seek some other theory. > > If some other theory is consistent, then it has a countable model. If a FIRST ORDER theory of predicate logic HAS AN INFINITE model, then it has a countable model. > > But ZFC is not > > inconsistent simply for Skolem's paradox nor is ZFC even rendered > > prohibitively counterintutitive by Skolem's paradox. > > That's why Skolem liked ZFC soo much? Yes, Skolem was critical of set theory. > > There exists a > > bijection from the universe of the countable model onto N; but that > > bijection is not itself a member of the universe of the countable > > model. And why should we assume that the universe of such a countable > > model must have such a function as a member? > > Because the bijection from the model (N') onto N *is* a countable set. > And if the model contains countable sets (N'), why then does it not > contain this set, which bijects N' and N? That's ridiculous reasoning. Just because a set has as members certain countable sets doesn't demand that the set have as members all countable sets. It would help if you knew just what a structure for a language is. > 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? MoeBlee
From: MoeBlee on 22 Nov 2006 13:48 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > The existence of a cardinal which bijects to a set S is not a necessary > > condition for the identity function on S to exist. Every set, whether > > well orderable or not, has an identity function, at least in such > > relatively sane systems as ZF and NBG. > > Is that an axiom? No. It is but an unjustified assumption. No, you ignoramus, it's a THEOREM from the axioms. > > > In order to assign a cardinal, at least one > > > bijection to an ordinal is required. > > > > If one does not assume the axiom of choice, as in ZF, then there ae sets > > without the sort of cardinality that WM requires, but such sets still > > can have identity functions on them, and in ZF, must have identity > > functions on them. > > No. No, you're ignorant. It is a theorem of Z set theory that for all x the identity function on x exists. MoeBlee
From: MoeBlee on 22 Nov 2006 14:10 mueckenh(a)rz.fh-augsburg.de wrote: > Countable means: There exists a bijection with N. After having > recognized that this simple definition leads to a contradiction, We've been over this and over this and over this, and you still persist incorrectly. You've not shown a statement P in the language of a Z set theory such that both P and ~P are theorems of the theory. > people > defined countable in and outside differently. I don't know who does that. I don't. > And if they will > recognize that there is really a contradiction in ZFC, then they will > define contradiction differently. I am sure, they will not fail. If there is shown a contradiction in ZFC, then I won't change any definitions; I'll just recognize that ZFC is inconsistent. But you've not shown a contradiction in ZFC. MoeBlee
From: Lester Zick on 22 Nov 2006 14:29
On 21 Nov 2006 16:13:20 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> I >> assume that part of my mathematics belongs to absolute truth which does >> not rely on arbitrary axioms. > >And you do not formulate your absolute truths as axioms, so your >absolute truths do not have objective form by which other people (not >just you) can verify that a given statement is of one of your absolute >truths. Your absolute truths are spoken by you only as verbiage that is >at worst vague and at best unformalized. You provide no objective means >by which other people (not just you) can verify that your arguments do >indeed follow by a specified logic from axioms which assert your >absolute truths. In this situation, all discussions can only defer to >you personally, as only you personally can say what is or is not >absolute truth as you have determined it and as you argue as to its >implications by your unspecified logic. > >On the other hand, even if you disagree that certain mathematical >axioms are true, at least they are given in utterly objective form as >formulas in a formal language and such that at least in principle one >can mechanically check whether a given sequence of formulas is or is >not a proof in a given formal system (and for the most basic theorems, >even in practice it wouldn't be difficult to fully formalize for >mechanical checkability). In other words, at least mathematics can give >you - up front, clearly, and precisely - exactly what its axioms are >and exactly by which rules theorems may be proven, while your polemics >offer no such intellectual courtesy or benefits. Well, Moe, you know I can completely agree with these comments. On the whole they represent a reasonably accurate assessment of the status quo in mathematics And I agree with your observations concerning explicit and objective form. However nothing in explicit and objective form guarantees a correct and accurate assessment of the implicit form and implications of one objective form in relation to others because that assessment is mechanized subjectively and not objectively. In other words you have objective axioms A, B, C, etc. which summarize various properties and predicates believed to be true explicitly. But you still have no way to evaluate the consistency among implications of A, B, C, etc. in relation to one another because those implications are not stated explicitly. ~v~~ |