An uncountable countable set
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From: Virgil on 4 Sep 2006 15:21 In article <1157367467.816725.158560(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Cantor's list has as many lines as my tree. But not as many "lines" as MY tree has paths. Again, the second proof was *not* about the reals. > > Please spare your nonsense. Cantor did not consider anything else than > the reals. Cantor's "diagonal" proof did not even concern them. It was others who later applied it to the set of reals. it was originally about the set of all lists (functions with domain N) or strings from an "alphabet" of two "letters". If one considers the alphabet of {"L","R"} for left branch and right branch, Cantor's original proof essentially proves that the number of paths in an infinite binary tree is uncountable. > Give my a tree of infinite paths consisting of 0's and 1's, and I show > that there > are not less edges than paths. Cantor shows less edges than paths. In a choice between a proof by Cantor and a proof by "Mueckenh", I will choose Cantor every time. > Give up your arguing. All set theorists know that the edges of my tree > are countable. Even Virgil knows it. But everyone knows that the set of paths is uncountable. Even Cantor has proved it.
From: Virgil on 4 Sep 2006 15:29 In article <1157367591.092721.79760(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > In current set theory there is no potential infinity at all. Ask > Virgil. If by "potential infinity" "Mueckenh" means "potentially infinite sets" then ,in set theory from ZF up, there is no distinction between "potential" versus "actual". Sets are either completely infinite or completely finite.
From: Virgil on 4 Sep 2006 15:34 In article <1157368118.735322.299590(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > I was not aware that abstract mathematical concepts (e.g., sets) > > had any physical constraints. > > Then you should learn it. It you are unable to physically (i.e. in > written form or in your mind) distinguish all the elements of a set, > then the set does not exist. While "in one's brain" may be considered physical, "in one's mind" is distinctly not physical. So that sets "exist in the mind" with no physical constraints whatsoever.
From: Dik T. Winter on 4 Sep 2006 19:40 In article <virgil-8D882D.13064304092006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > But if you wish, provide a definition of "number". As far as I know, > > > there is not one in mathematics. > > > > A number has only one of the following properties: It is larger than or > > smaller than or equal to any natural number. > > A complex number is a number which need not have any of these properties. > > And before one uses "larger", "smaller" or "equal", in any definition, > one must define them too. Moreover, I see now it is also badly worded. As it stands, I think, are only three properties listed: (1) larger than any natural number (2) smaller than any natural number (3) equal to any natural number. And a number should satisfy only one of them. Question, is 3 a number? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 4 Sep 2006 20:02
In article <virgil-6E61C8.13295404092006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > In article <1157367591.092721.79760(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > In current set theory there is no potential infinity at all. Ask > > Virgil. > > If by "potential infinity" "Mueckenh" means "potentially infinite sets" > then ,in set theory from ZF up, there is no distinction between > "potential" versus "actual". Sets are either completely infinite or > completely finite. Before we can discuss this in a correct mathematical sense, we have first to have a mathematical definition of "potential infinite" vs. "actual infinite". I think a proper distinction can be made between the contents of a set and the set itself. If we have an initial segment of the ordinals, we can state that the ordinal number is some alpha, and that clearly can be infinite, and in that case the set is actually alpha. On the other hand, we can define that the contents are potentially alpha. So the set {0, 1} has actual ordinal 2 and the contents have potential ordinal 2. In the case that alpha is not a limit ordinal this would mean that the largest ordinal in the set is the predecessor of alpha. If it is a limit ordinal, there is no largest ordinal in the set. With this definition, N has ordinal w, which is an actual infinity, but the contents are potentially w, so they are of a potential infinity. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |