Cantor Confusion
in [Math]
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From: William Hughes on 3 May 2007 10:05 On May 3, 8:50 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 3 Mai, 13:18, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 3, 7:12 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > Wrong. If there is always a path p' with p, then for every node of p > > > we have a path p' such that K(p, n) = K(p', n). > > > Look over there! A pink elephant! > > > p' never changes. > > > >Hence p is never > > > different from every other path. > > > since there is a single fixed path, p', which is not > > the same as p. > > 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. - William Hughes
From: William Hughes on 3 May 2007 10:13 On May 3, 6:43 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 30 Apr., 19:28, Virgil <vir...(a)comcast.net> wrote: >> > > The ZF system and the NBG system, for example, both not only allow, they > > require that CIBTs exist and that the set of paths in them are > > uncountable. > > That is their problem: These systems are self-contradictory. > Only in Wolkenmuekenheim Ouside of Wolkenmukenheim the two statetments A: p is never the only path in a path bundle B: if p' is not equal to p, then there is a path bundle than contains p but not p' are both true, although Muekenheim does not like the fact that statement B can be true even though statement A is true. Inside of Wolkenmuekenheim, A and B are contradictory. - William Hughes
From: MoeBlee on 3 May 2007 12:13 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. There are axioms (of which it is entirely mechanical to check whether a formula is or is not an axiom) and rules of inference (of which it is entirely mechancial to check whether a purported application of a rule is indeed an application of the rule). And a proof (a sequence of formulas, each of which is an axiom or follows from previous formulas in the sequence by an inference rule) in formal Z set theory of the uncountability of the reals exists. > > If you understand the basic > > material, there is not much to believing, since it is a simple of > > matter of observing that a certain sequence of formulas that satisfies > > the definition of 'first order proof from the Z axioms' does exist. > > And as easily one can see that the difference between number of paths > and number of nodes in the binary tree is given by > 2-1-1+2-1-1+2-1-1+-... which is not much more than 2. If you > understand the basic material, there is not much to believing. So it's still unclear whether you do understand that if you just studied the basic material you too would observe that there is a proof in formal Z set theory, with no implicit assumptions, that the set of real numers is uncountable. MoeBlee
From: Virgil on 3 May 2007 13:42 In article <1178189003.748513.79940(a)l77g2000hsb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 30 Apr., 19:28, Virgil <vir...(a)comcast.net> wrote: > > In article <1177936379.128093.174...(a)n76g2000hsh.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > Anyhow. The number of path bunches surpassing the number of nodes is > > > zero. If any infinite path p exists, then the bunch {p} does exist > > > too, and {p} is a path bunch. If, on the other hand, {p} does not > > > exist, then it is ridiculous to try to count the number of paths. > > > > So that WM is essentially saying that there is no such thing as an > > infinite binary tree. > > No meaningful calculating with infinite numbers is possible without > such contradictions arising. That WM claims things that can be proved false in ZF and NBG, is WM's problem, not the problem of mathematics. > > > > The problem is that he has no axiom system in which he can prove his > > conclusion whereas we have several in which we can prove that such trees > > do exist and have uncountably many paths in them. > > > > The ZF system and the NBG system, for example, both not only allow, they > > require that CIBTs exist and that the set of paths in them are > > uncountable. > > That is their problem: These systems are self-contradictory. Claimed but not proven. That they conflict with WM's religious faith does not make them self-contradictory. > > > > What axiom system does WM claim allows him to prove otherwise? > > > 2-1-1+2-1-1+2-1-1+-... is never much more than 2. That hardly constitutes an axiom system, and does not show that ZF or NBG are anything but consistent.
From: Virgil on 3 May 2007 13:47
In article <1178189210.101526.172170(a)p77g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 30 Apr., 19:33, Virgil <vir...(a)comcast.net> wrote: > > In article <1177936511.069913.212...(a)e65g2000hsc.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 27 Apr., 22:06, Virgil <vir...(a)comcast.net> wrote: > > > > > > > If something is valid for all elements, then it does not change if the > > > > > number of elements is infinite. > > > > > > What is true of elements one at a time need not be true for infinite > > > > sets of them. > > > > I.e., properties of members of a set need not be inherited by the sets > > themselves. > > > > > Why then do you believe that Cantor's diagonal proof is true? > > > > Cantor's proof does not say anything is true for a set itself, only for > > each member of a set, so it is not the same thing. > > 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. |