Cantor Confusion
in [Math]
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From: William Hughes on 11 May 2007 07:12 On May 11, 4:21 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: gument. > > Ad infinitum there is a path p' with p, because ad infinitum there is > no node where p is single. Therefore, ad infinitum, there is a path > which shares with p the node in question and all earlier nodes too. Look! Over There! A pink elephant! ad infinitum it is always the same path. -William Hughes
From: William Hughes on 11 May 2007 07:31 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. Look! Over there! A pink elephant! The first node at which another path is required does not exist. - William Hughes
From: William Hughes on 11 May 2007 07:42 On May 11, 5:35 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 11 Mai, 00:25, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On May 10, 6:32 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 9 Mai, 22:43, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > >...there must be such a path. > > > > >Why not call it p''. > > > > > because the name p'' has already been used > > > > for a path that does branch off from p. > > > > No, it has not. p'' is, by definition, that path which is with p also > > > at the next node, wherever you rest. > > > Funny, I thought it was different. > > No. That was my original definition That was my original definition (Whoops, someone else gave a definition before me) Look! Over there! A pink elephant! That was the original defintion. - William Hughes
From: WM on 11 May 2007 08:31 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. 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). > > Look! Over there! A pink elephant! There is a node where p* changes. Regards, WM
From: William Hughes on 11 May 2007 08:36
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! Remark: When you change n, p* never has to change. - William Hughes |