Cantor Confusion
in [Math]
|
From: Lester Zick on 10 Feb 2007 12:28 On Sat, 10 Feb 2007 12:06:02 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> On 9 Feb., 21:26, "MoeBlee" <jazzm...(a)hotmail.com> wrote: >> >> > > Reality. >> > >> > What reality? Empirically testable physical reality? A mathematical >> > statement is not of that kind. >> >> Every mathematical statement is of that kind. You would recognize this >> (if you could, but you couldn't) if all reality ceased to exist. > >Are all statements (not just mathematical ones) about reality? Sure. Why not? Perhaps you'd care to suggest a statement that is not about reality? >> No >> mathematics would remain. Yes, to grasp these facts, without such a >> bad experience, i.e., during the continued existence of reality, one >> needs some deeper thinking than is required to understand the meaning >> of some crazy axioms. ~v~~
From: Fuckwit on 10 Feb 2007 12:36 On Sat, 10 Feb 2007 10:28:09 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: > > Sure. Why not? Perhaps you'd care to suggest a statement that is not > about reality? > 0 e IN.
From: mueckenh on 10 Feb 2007 12:51 On 10 Feb., 12:13, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: > > >> > Instead of a path P consider the set S of nodes K which belong to a > >> > path P. Do your calculation and arguing. Then substitute P for S to > >> > have a brief notation. > > >> One cannot determine merely from a set of nodes for a given tree > >> whether that set of nodes is or is not a path. > > > > One can. > > Then please do so: > > given tree T := { a, b, c, d, e, f, g } > given set of nodes S := { a, b, c } > > Tell us whether S is a path in T. And please explain that. Pardon, I overlooked your first question. The tree T(2) in my notation can be given as a chain by a bc gfed abc is the subtree T(1) with only (the root node and) one level. It is not a path. Here are some further representations of the tree T(2): 0. | \ 0 1 | \ | \ 0101 briefly (edges are not necessary) 0. 0 1 0101 (0,1) (1,1) (1,2) (2,1) (2,2) (2,3) (2,4) The last is the coordinate representation. I think it is obvious which subsets are paths. One of the paths in the infinite tree is the set of nodes p(oo) = {(n, 1)| n in N or n = 0}. It can also be written as a,b,g, ... or as 0.000.... In no case the order property need be mentioned. Regards, WM
From: Franziska Neugebauer on 10 Feb 2007 13:39 mueckenh(a)rz.fh-augsburg.de wrote: > On 10 Feb., 12:13, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: >> mueck...(a)rz.fh-augsburg.de wrote: >> > On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: >> >> >> > Instead of a path P consider the set S of nodes K which belong >> >> > to a path P. Do your calculation and arguing. Then substitute P >> >> > for S to have a brief notation. >> >> >> One cannot determine merely from a set of nodes for a given tree >> >> whether that set of nodes is or is not a path. > >> >> > One can. >> >> Then please do so: >> >> given tree T := { a, b, c, d, e, f, g } >> given set of nodes S := { a, b, c } >> >> Tell us whether S is a path in T. And please explain that. > > Pardon, I overlooked your first question. > > The tree T(2) in my notation can be given as a chain by > > a > bc > gfed > > abc is the subtree T(1) with only (the root node and) one level. It is > not a path. Strange. My version of the tree T a / \ b d / \ / \ c e f g obviously contains a path having (the node set) S = { a, b, c }. Hence the attempt to *uniquely* represent trees and/or paths by plain sets of nodes fails. F. N. -- xyz
From: David Marcus on 10 Feb 2007 15:36
Lester Zick wrote: > On Sat, 10 Feb 2007 12:06:02 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >mueckenh(a)rz.fh-augsburg.de wrote: > >> On 9 Feb., 21:26, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > >> > >> > > Reality. > >> > > >> > What reality? Empirically testable physical reality? A mathematical > >> > statement is not of that kind. > >> > >> Every mathematical statement is of that kind. You would recognize this > >> (if you could, but you couldn't) if all reality ceased to exist. > > > >Are all statements (not just mathematical ones) about reality? Something from the "Lord of the Rings" trilogy. -- David Marcus |