Cantor Confusion
in [Math]
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From: WM on 5 May 2007 08:00 On 3 Mai, 19:47, Virgil <vir...(a)comcast.net> wrote: > > Cantor's proof says that all members of the set R are more than all > > members of the set N. > > In the sense that there is an injection from N to R but none from R to > N, right! > > > This is a wrong conclusion. > > It is a validly proved conclusion in ZF and NBG, and WM has not come up > with any axiom system in which he can prove it false. > > And as far as mathematics is concerned, that settles the issue. Definition: A set M is countable, if there is a one-to-one correspondence with the set N of natural numbers. There is a one-to-one correspondence of M and N if and only if all elements m_n of M can be enumerated, i.e., if and only if a list can be defined: 1) m_1 2) m_2 3) m_3 .... n) m_n .... Theorem 1: The set of finitely definable numbers is countable and can be put into a list. Proof: All finite definitions are finite words over a finite alphabet. All finite words can be ordered lexicographically. This is the finite definition of a list. Theorem 2: The set of finitely defined numbers cannot be put into a list. Proof: Assume such a list exists. This means, such a list has been finitely defined. Construct the diagonal number. It is a finitely defined number. Find the set of finitely definable numbers is uncountable. So there are countable sets which are not in one-to-one correspondence with the set of natural numbers? Regards, WM
From: William Hughes on 5 May 2007 13:09 On May 5, 7:42 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 3 Mai, 16:05, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > At every node K(p, n) of p there is a single fixede path p', which > > > has been with p for all nodes K(p, m < n). Yes, without pink > > > elephants, we can find at every node K(p, n) such a path p' > > > Look over there! A pink elephant! > > > p' never changes when n changes. > > > >So p is never single. > > > However, it is not necessary for p to be single for p to be separated > > from > > a path p'. It is possible for p to be separated from every p' without > > p ever being single. > > That is obviously nonsense for linear sets which, after having been > separated, never meet again. Thats right, after the last path p' has separated p has to be single. Whoops, how does one show there is a last path. Look! Over there! A pink elephant! - William Hughes
From: Carsten Schultz on 5 May 2007 13:46 WM schrieb: > On 3 Mai, 19:47, Virgil <vir...(a)comcast.net> wrote: > >>> Cantor's proof says that all members of the set R are more than all >>> members of the set N. >> In the sense that there is an injection from N to R but none from R to >> N, right! >> >>> This is a wrong conclusion. >> It is a validly proved conclusion in ZF and NBG, and WM has not come up >> with any axiom system in which he can prove it false. >> >> And as far as mathematics is concerned, that settles the issue. > > Definition: A set M is countable, if there is a one-to-one > correspondence with the set N of natural numbers. > There is a one-to-one correspondence of M and N if and only if all > elements m_n of M can be enumerated, i.e., if and only if a list can > be defined: > > 1) m_1 > 2) m_2 > 3) m_3 > ... > n) m_n > ... > > Theorem 1: The set of finitely definable numbers is countable and can > be put into a list. > > Proof: All finite definitions are finite words over a finite alphabet. > All finite words can be ordered lexicographically. This is the finite > definition of a list. > > Theorem 2: The set of finitely defined numbers cannot be put into a > list. > > Proof: Assume such a list exists. This means, such a list has been > finitely defined. It has been explained a lot of times to Mückenheim that this is not a valid conclusion. He is just a liar. > Construct the diagonal number. It is a finitely > defined number. Find the set of finitely definable numbers is > uncountable. > > So there are countable sets which are not in one-to-one correspondence > with the set of natural numbers? > > Regards, WM > -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page.
From: Virgil on 5 May 2007 14:36 In article <1178365354.115147.180260(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 3 Mai, 16:05, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > At every node K(p, n) of p there is a single fixede path p', which > > > has been with p for all nodes K(p, m < n). If K(p,n) refers to the nth node of path p, then in every CIBT, there are uncountably many paths which have "been with" p for all nodes K(p, m) with m < n, for any n. The only way to "separate" one path of CIBT from all others is to take an infinite subset of its nodes, through which there will be only one path, Through every node in any finite set of nodes, either no path passes or uncountably many pass. > > No, p is never separated from those uncountably many paths which you > cannot ask for One does not have to ask for what is already there.
From: Virgil on 5 May 2007 14:50
In article <1178366048.843043.67310(a)n76g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 3 Mai, 18:13, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On May 3, 3:55 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 30 Apr., 20:04, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > > On Apr 30, 5:35 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > On 27 Apr., 22:06, Virgil <vir...(a)comcast.net> wrote: > > > > > Why then do you believe that Cantor's diagonal proof is true? > > > > > > You'd have to define 'true proof'. Meanwhile, with just some basic > > > > knowledge of predicate calculus and set theory, one can see that > > > > Cantor's diagonal argument is formalizable into a proof in first order > > > > logic from the axioms of Z set theory. > > > > > There is an implicit assumption entering which is wrong. > > > The difference of 1 between the digits of two numbers is insufficient > > > to distinguish these numbers in the limit n --> oo. > > > lim(n-->oo) 10^-n = 0. Only this limit makes power series of digits > > > converge and makes real numbers exist. But Cantor's proof forgets > > > that. > > > > There are no implicit assumptions in formal Z set theory. > > > What you say proves that you never have thought about that topic deep > enough. Not at all. it is just that all assumptions are explicit, so no implicit ones are needed, or used. > One implicit assumption is that, in Hessenberg's proof of > Cantor's theorem, the set of all natural numbers which are not mapped > on subsets of P(N) which do contain them, does exist The existence of that set follows directly from the axiom schema of specification in ZF. > > You are wrong to assume that I did not study set theory. But the > formalism veils the problem it is built upon. The formalism is all there is. > > > in formal Z set theory, with no implicit assumptions, that the set of > > real numers is uncountable. > > Even if this was clear, then the result > 2-1-1+2-1-1+2-1-1+-... <= 2 > would show that set theory is self contradictive. Only to such True Believers as choose not to see that they are assuming things not true in ZZF or NBG. |