Cantor Confusion
in [Math]
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From: William Hughes on 22 Apr 2007 13:07 On Apr 22, 9:08 am, mueck...(a)rz.fh-augsburg.de wrote: > On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: <snip> > > Is the following statement true of false > > > There are enough nodes to separate an uncountable number > > of paths. > > Definitively: False. > Recall M: a set of separation points can separate a path from all other M: paths. Which of these statements is false? A path can be separated from all other paths by a set of nodes There are uncountably many sets of nodes - William Hughes
From: mueckenh on 24 Apr 2007 09:21 On 22 Apr., 19:07, William Hughes <wpihug...(a)hotmail.com> wrote: > On Apr 22, 9:08 am, mueck...(a)rz.fh-augsburg.de wrote: > > > On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > <snip> > > > > Is the following statement true of false > > > > There are enough nodes to separate an uncountable number > > > of paths. > > > Definitively: False. > > Recall > > M: a set of separation points can separate a path from all other > M: paths. > > Which of these statements is false? > > A path can be separated from all other paths by a set of nodes > > There are uncountably many sets of nodes False is the implication that n nodes can separate more than n+1 path bunches (where a path is a path bunch with one element). Regards, WM
From: William Hughes on 24 Apr 2007 12:20 On Apr 24, 9:21 am, mueck...(a)rz.fh-augsburg.de wrote: > On 22 Apr., 19:07, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Apr 22, 9:08 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > > <snip> > > > > > Is the following statement true of false > > > > > There are enough nodes to separate an uncountable number > > > > of paths. > > > > Definitively: False. > > > Recall > > > M: a set of separation points can separate a path from all other > > M: paths. > > > Which of these statements is false? > > > A path can be separated from all other paths by a set of nodes > > > There are uncountably many sets of nodes <Snip failure to address the question.> Try again Which of these statements is false? A path can be separated from all other paths by a set of nodes There are uncountably many sets of nodes - William Hughes
From: William Hughes on 24 Apr 2007 12:23 On Apr 24, 9:21 am, mueck...(a)rz.fh-augsburg.de wrote: > On 22 Apr., 19:07, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Apr 22, 9:08 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > > <snip> > > > > > Is the following statement true of false > > > > > There are enough nodes to separate an uncountable number > > > > of paths. > > > > Definitively: False. > > > Recall > > > M: a set of separation points can separate a path from all other > > M: paths. > > > Which of these statements is false? > > > A path can be separated from all other paths by a set of nodes > > > There are uncountably many sets of nodes > <snip failure to address the question> Try again Which of these statements is false? A path can be separated from all other paths by a set of nodes There are uncountably many sets of nodes - William Hughes
From: mueckenh on 24 Apr 2007 12:46
On 22 Apr., 18:48, Virgil <vir...(a)comcast.net> wrote: > In article <1177234322.313795.125...(a)l77g2000hsb.googlegroups.com>, > > mueck...(a)rz.fh-augsburg.de wrote: > > On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > > Each of these bunches contains an uncountable number > > > of paths. All you show is that there are a countable number > > > of bundles, each of which contains an uncountable number of paths. > > > What about Cantor's list? Does it contain the representations of paths > > or of path bundles? > > Is a "path bundle" a set of paths? If so why not call it a set of paths? > > > If your answer is "paths", what is the difference > > to a path bunch with one element? > > What is the difference between a "path bundle and a "path bunch"? I use bunch, Dik uses bundle. I see no difference. > > > > Each of these paths can be separated from all other paths by > > > a set of nodes. There are uncountable many sets of nodes. > > > There are enough nodes to separate uncountable many paths. > > > Your error lies in the fact, that one node can only once be used to > > separate two path bunches. > > False! There are sets of paths which cannot be isolated from each other > by any one node. Which sets do you have in mind? > > > Therefore the "sets of nodes" (which are > > nothing but paths) do not help to show the uncountability of paths. > > You are lacking some logic (quite a lot). You try to prove the > > uncountability of a set by the uncountability of that set. > > False! > > We succeed in proving uncountability of the set of paths of a CIBT by > proving that the set of paths bijects with the uncountable power set of > the set of levels of a CIBT. That is nonsense. These uncountability proofs are wrong. Further, if they were correct, I show a contradiction in set theory. This cannot be rebutted by a different proof. Regards, WM |