Cantor Confusion
in [Math]
|
From: William Hughes on 11 May 2007 09:20 On May 11, 8:31 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 11 Mai, 13:31, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 11, 4:57 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 10 Mai, 23:03, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > Note: you can enumerate an infinite set of nodes. > > > > Let us assume that. > > > > > For every node n belonging to the set of nodes that you can enumerate, > > > > one path p'(n) is sufficient to accompany p up to that node. > > > > p'(n) is not the same for every node. > > > > For which node is another path required? > > > p(1) is not identical to p. Thus there exists a node, > > call it M_1, where p and p(1) branch in different directions. > > Thus, the first node at which another path is requrired > > is node M_1+1. > > > The first node at which another path is required depends on the > > choice of p(1) and what is more, given k equal to any natural number > > greater than 1, you can choose p(1) to have M_1+1 = k. > > What is more, given p(1), you can compute M_1+1, and find > > another path, p'(1), with M_1'+1 > M_1 +1. > > What is more, given p'(1), you can find > > another path, p''(1), with M_1''+1 > M_1' +1. > > What is more, given p''(1), you can find > > another path, p'''(1), with M_1'''+1 > M_1'' +1. > > You can continue this, ad infinitum, however, > > at each step you have a first node at which a different > > path is required. > Deciding not to respond directly to this, Muekenhiem simply asserts: > And at each node you have a path p* with p that is the same from the > beginning and remains so forever (if there is a forever). > > At each node you get a path that is with p up to that node. However, at no node do you get a path that remains with p forever. (p" is not equal to p) So there must be some node at which p" changes. And only in Wolkenmuekenhiem is something that changes a path. - William Hughes.
From: William Hughes on 11 May 2007 09:48 On May 11, 2:54 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > This is > contradicted by the fact that every node of p which you can ask for is > shared by the same path p'. > Up to now we have been talking about the set "every node of p". You have now introduced the Wolkenmuekenheim set "every node of p which you can ask for". This set is bounded above by the maximum node. Of course the maximum node can change to something bigger (this makes sense in Wolkenmuekenheim) and when it does the set "every node of p which you can ask for" changes. So the set "every node of p which you can ask for" can change. When it does the "same path p' " may have to change. - William Hughes
From: Aatu Koskensilta on 11 May 2007 10:28 On 2007-05-11, in sci.math, William Hughes wrote: > Look! Over there! A pink elephant! > > That was the original defintion. I wouldn't know about that, but it's certainly a very original definition. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: WM on 11 May 2007 11:46 On 11 Mai, 14:36, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 11, 5:35 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > Definition: p* is a path in the binary tree which is with path p at > > the next node K(p, n+1) in the binary tree, whatever node K(p, n) you > > investigate. > > > Lemma: In the binary tree for every path p there exists at least one > > path p* as defined. > > > Proof: Choose a node K(p, n) of path p in the binary tree. There is a > > path p* which is with p at node K(p, n+1) too such that K(p, n+1) = > > K(p*, n+1). This path p* has been with p at all nodes m < n too. The > > choice of n is arbitrary. Therefore the Lemma holds for all nodes K(p, > > n) of p with n in N. > > > Remark: This proof is independent of the number of nodes. > > Look! Over there! A pink elephant! You should consult an eye doctor. > > Remark: When you change n, p* never has to change. It cannot change? I don't fix it. Regards, WM
From: WM on 11 May 2007 11:53
On 11 Mai, 15:20, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 11, 8:31 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 11 Mai, 13:31, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On May 11, 4:57 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 10 Mai, 23:03, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > Note: you can enumerate an infinite set of nodes. > > > > > Let us assume that. > > > > > > For every node n belonging to the set of nodes that you can enumerate, > > > > > one path p'(n) is sufficient to accompany p up to that node. > > > > > p'(n) is not the same for every node. > > > > > For which node is another path required? > > > > p(1) is not identical to p. Thus there exists a node, > > > call it M_1, where p and p(1) branch in different directions. > > > Thus, the first node at which another path is requrired > > > is node M_1+1. > > > > The first node at which another path is required depends on the > > > choice of p(1) and what is more, given k equal to any natural number > > > greater than 1, you can choose p(1) to have M_1+1 = k. > > > What is more, given p(1), you can compute M_1+1, and find > > > another path, p'(1), with M_1'+1 > M_1 +1. > > > What is more, given p'(1), you can find > > > another path, p''(1), with M_1''+1 > M_1' +1. > > > What is more, given p''(1), you can find > > > another path, p'''(1), with M_1'''+1 > M_1'' +1. > > > You can continue this, ad infinitum, however, > > > at each step you have a first node at which a different > > > path is required. > > Deciding not to respond directly to this, Muekenhiem > simply asserts: > > > And at each node you have a path p* with p that is the same from the > > beginning and remains so forever (if there is a forever). > > At each node you get a path that is with p up to that node. Why should I use that path for any applications? > However, at no node do you get a path that remains with p forever. Then there must be a node wat which p is single. Otherwise from all the nodes at which p is not single, we form a sequence of nodes which is p*. > (p" is not equal to p) > So there must be some node at which p" changes. p* is that path which consists of all the nodes at which p is not single (so that there is also another path). Is it impossible or forbidden by ZFC or third order logic or pink elephants to construct a path p* from this set of nodes? Which nodes are available for path construction? Regards, WM |