Cantor Confusion
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From: mueckenh on 30 Nov 2006 03:44 Eckard Blumschein schrieb: > 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? Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations of Set Theory", 2nd edn., North Holland, Amsterdam (1984) > > > Then we have finite sets without > > a largest element. > > Why not with a growing largest element? That depends on definition. The largest element of a set of numbers today is not the same numbers as the largest element of the set tomorrow. But the object "largest element" considered as a variable, in fact would grow. Nevertheless this is not what Fraenkel et al. wish to express. They talk about the development of the set of all sets in a Platonic world view. Regards, WM
From: mueckenh on 30 Nov 2006 03:49 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 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. > > I don't know what you mean by 'demonstrated for few symbols'. > > Meanwhile, there does exist a proof in Z set theory that there exists a > bijection between the set of natural numbers and the set of ordered > pairs of natural numbers. Your "proof" concerns less than 10^100 elements. You are not even able to express some natural numbers of this domain, but you insist that there was a proof concerning all of them. A ridiculous self-overestimation. Regards, WM
From: mueckenh on 30 Nov 2006 03:53 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > No. We have to accept that there are sets which are capable of > > growing, as Fraenkel et al. express it. Then we have finite sets without > > a largest element. Then we describe reality correctly. > > You're a dishonest fool. If you continue to loose your self control in this way, then I will have to cease discussing with you. > I've already explained to you, as you should > have read for yourself in the original source, that what Fraenkel, > Bar-Hillel, and Levy mean by the universe of sets "growing" (THEIR > scare quotes) is that different axioms yield different universes of > sets, That is nonsense. I know that you cannot recognize it. Only for the lurkers: If *different* sets generated by different axiom systems were meant, then Fraenkel et al. would not only have to talk about "growing" but also about "shrinking" or simply about differing sets. But they don't. > not that any given set itself grows, let alone that there are > finite sets without a largest element (!). You misunderstand Fraenkel and you misunderstand me. Small wonder. I did not say that Frankel talks about these growing sets with the same meaning as I do. Please read carefully (try it for one time). I wrote: " We have to accept that there are sets which are capable of growing, *as Fraenkel et al. express it*. Regards, WM
From: mueckenh on 30 Nov 2006 03:55 MoeBlee schrieb: > 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. I did not say that I had proven that. I said "In particular because there is neither a standard model nor a non-standard model of ZFC." Are you really incapable of understanding such simple sentences? > > > > > 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. Do you really believe that set theory becomes more popular if its adherents turn out to behave like undisciplined children without any self-control like you or Mr Bader? Regards, WM
From: mueckenh on 30 Nov 2006 04:01
Virgil schrieb: > > > > 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. I did. Take the natural numbers between [e*10^10^100] and [pi*10^10^100]. You do not even know how many there are. But you insist to prove something for them all. > > Particularly when WM acknowledges himself not to be a mathematician, and > mathematicians claim otherwise? I am not a set theorist. > > 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. Unfortunately today "mathematically valid" is frequently mistaken with "matheologically believed". Regards, WM |