Cantor Confusion
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From: MoeBlee on 29 Nov 2006 13:30 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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? As I said, you've not proven that there is no model of ZFC. > > > 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! WRONG. We DO. You're an ignoramus. MoeBlee
From: Virgil on 29 Nov 2006 13:34 In article <1164817491.026348.109040(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? > > As it is grammatically not an assertion, not even a complete sentence, yes! > > 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? If one is talking about the same induction as Peano's, then no finite sets are generated by that form of induction. > > > > 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. That particular corruption does not, and cannot, occur within any standard set theory. > Then we have finite sets without > a largest element. Then we describe reality correctly. Since WM's notion of "reality" is irrelevant to mathematics, it can do without WM's unsettled sets quite nicely. > > Regards, WM
From: MoeBlee on 29 Nov 2006 13:41 Eckard Blumschein wrote: > On 11/29/2006 5:24 PM, mueckenh(a)rz.fh-augsburg.de wrote: > We have to accept thart there are sets which are capable of > > growing, as Fraenkel et al. expess it. > > While there are several books by Fraenkel: > > Einleitung in die Mengenlehre 2nd ed. 1923, 3rd ed. 1946, > Gesammelte Abhandlungen Cantor, Dedekind 1932 > Das Leben Georg Cantors 1932 > Abstract set theory 1961 > Lebenskreise - Aus den Erinnerungen eines jüdischen Mathematikers 1967 > > perhaps there is only one book by Fraenkel et al.: > Foundations of Set Theory 1958. Correct? > > > Then we have finite sets without > > a largest element. > > Why not with a growing largest element? Because Fraenkel, Bar-Hillel, and Levy said nothing that implies any such thing as that there are finite sets without a largest element. WM is just latching onto a particular quote(that even uses scare quotes) out of context and with utter disregard to what the quote is meant to actually summarize. MoeBlee
From: Virgil on 29 Nov 2006 13:41 In article <1164817781.982393.126010(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. If it is not a proof, or rather if the bijection does not exist in ZF of NBG or same other set theory, then WM should be able to produce a counterexample. Absent any evidence stronger that WM's assurances that no such bijection exists, why should we believe him. Particularly when WM acknowledges himself not to be a mathematician, and mathematicians claim otherwise? WM does not get it. In mathematics, claims unsupported by mathematically valid proofs do not persuade, particularly when opposed by mathematically valid proofs of their falsehood.
From: Virgil on 29 Nov 2006 13:45
In article <1164818051.564389.180950(a)14g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? That such a model does not exist in WM's mind is not evidence that it does not exist in minds competent to comprehend it. > > > > > 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! One has. But defining something does not instanciate it. One can define square circles and 4 sided triangles, too. > > Regards, WM |