Cantor Confusion
in [Math]
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From: Virgil on 3 May 2007 13:57 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. What "contradictions"? That WM says things he cannot prove does not make for contradictions with things that ZF or NBG says which have been proved. > > > > 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. They may contradict WM but that does not make them self-contradictory. Or does WM think himself able to speak ex cathedra? > > > > 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. Irrelevant to the existence of bijections between the power set of N and the set of all paths of a CIBT. Which bijections prove WM wrong. > > Regards, WM
From: Virgil on 3 May 2007 14:03 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. > This is a wrong conclusion. Cantor's proof shows that there are "more" infinite binary sequences that naturals. A variation on this valid proof, not actually by Cantor, also shows that it holds for reals between 0 and 1 in any base representations with base > 3 (and,, with slight modifications, also for bases 2 and 3). > Only for each diagonal number one can say, > at most, that it was not in the list while it, nevertheless belongs to > a countable set. WM may say it, but that does not make it true, nor does WM have any mathematically valid proof of his claim. > > Regards, WM
From: Virgil on 3 May 2007 14:18 In article <1178189738.719781.210340(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(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. What WM is describing is not a difference between two fixed numbers but between two sequences of numbers whose differences converge to 0, the limiting difference is indeed zero but that is totally irrelevant, In base b, numbers of form m/b^n have two expansions. But excluding these, a difference of 1 or more in any digit position differentiates between two different reals. > 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. Since the original Cantor theorem was about infinite binary strings, each binary bit had a natural number index, but there were no place values at all involved, unless they were those natural number indices, so nothing "went to zero". Thus WM's "going to zero" argument is irrelevant to the actual issue. > > > 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. I do not believe in WM's "understanding" as I can, for example, clearly see a bijection between the set of all paths in a CIBT and the set of all subsets of N. That WM refuses to see what is so clear marks hi as one who desperately clings to his faith despite any evidence of its errors.
From: Virgil on 3 May 2007 14:34 In article <1178190762.081101.159970(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > Hence Cantor's diagonal proof fails. It has been clearly proved that the power set of any set cannot be injected into the set itself, but it is obvious that any set can be injected into its power set. Thus Card(P(N)) > Card(N). It has also been shown that there is a bijection between P(N) and the set all binary strings (mappings from N to, say, {w,m}). But Cantor's original diagonal proof merely stated that there was no injection from the set of such binary strings to N. So where does the proof fail? > Because p can never be distinguished from all other paths in the tree, > but only from those few (countably many) paths p' which you can ask > for. As the set of paths is essentially the set of binary strings, WM is wrong again. > > A severe terminology problem. Bunches do not split, they continue until > > they terminate. > > Bunches terminate by splitting into other bunches. If they would not > split, then they would not terminate. If a "bunch" is a set of paths, then how does a bunch "terminate" in a CIBT when none of its members do? > > Right. So there are infinitely many bunches of paths starting at the root > > node. And so bunches of paths do not split, they either terminate, or they > > continue. > > They terminate because they split. If, by "terminate" WM means that there is some last node that all the paths in that "bunch" have in common, then WM is limiting "bunch" to apply only to sets of paths having finitely many nodes in common, in which case every "bunch" has uncountably many paths in it. > > It is the number of separations which counts! Not in CIBTs. There the set of paths has the same cardinality as the power set of N.
From: Virgil on 3 May 2007 14:36
In article <1178190896.714858.73840(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 2 Mai, 20:15, Virgil <vir...(a)comcast.net> wrote: > > In article <1178106864.347366.36...(a)c35g2000hsg.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > >http://groups.google.nl/group/AOTI/browse_thread/thread/da0a93f42305b2c8 > > > > WM's "intercession" of two sets is merely having each set of two sets > > dense in their suitably ordered union, and does not eliminate the very > > real differences between sets incapable of being bijected. > > Very real? Real is reality, but no tansfinite beliefs. WM's faith in his religion is touching, but mathematics is about what follows from a set of axioms, not about faith unsupported by fact. |