An uncountable countable set
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From: Daryl McCullough on 23 Jun 2006 17:07 mueckenh(a)rz.fh-augsburg.de says... >Daryl McCullough schrieb: > >> No, that doesn't follow. What follows is that M(f) is *not* a subset >> of S. Since S contains all finite subsets of N, it follows that M(f) >> contains some element that is *not* a finite subset of N. > >Consider the mapping f : |N --> M, where M contains a countable set of >*infinite* subsets of |N and K. > >What follows now? If K(f) = { x in N | x is not an element of f(x) }, and M is some set containing K(f), then it follows that f is not a surjection from N to M. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 23 Jun 2006 17:18 mueckenh(a)rz.fh-augsburg.de says... >Daryl McCullough schrieb: > >> If an assumption leads to a contradiction, then that assumption must >> be false. > >The assumption includes that |N and P(|N) do exist as complete sets. Actually, the nonexistence of a mapping from N to P(N) has nothing to do with whether N and P(N) exist as "complete sets". If you want to think of them as potentially infinite, that's fine. It's still possible to have a bijection between two potentially infinite sets. The function f(x) = 2*x is a surjection from the set of naturals to the set of even naturals. It doesn't matter whether you think of the naturals as "completed" or not; if x is any natural whatsoever, I can multiply by 2 to get an even natural. And given any even natural, I can divide by two to get a natural. It doesn't matter whether the naturals are "complete" or not. Similarly, there is a surjection from P(N) to [0,1]. Given any set S of naturals, I can generate a real number as follows: r(S) = sum over all j in S of 2^{-j} I don't need to think of the set [0,1] as "completed" in order to understand that this mapping is surjective. -- Daryl McCullough Ithaca, NY
From: Richard Tobin on 23 Jun 2006 17:45 In article <e7h9gb$pst$1(a)nntp.aioe.org>, Dann Corbit <dcorbit(a)connx.com> wrote: ><mueckenh(a)rz.fh-augsburg.de> wrote in message >news:1151079756.995887.177600(a)p79g2000cwp.googlegroups.com... >[snip] >> It is ridiculous to believe in Copernicus in the frame work of the >> bible. Nevertheless I believe in Copernicus. >>What (exactly) is the place in the Bible that contradicts anything said by >Copernicus? Joshua 10:13 describes a miraculous event in which Joshua commands the sun to stand still, with the result that the day continues until the children of Israel had avenged themselves upon their enemies. This, it was argued, shows that the cycle of day and night is normally caused by the movement of the sun. -- Richard
From: Virgil on 23 Jun 2006 18:30 In article <e7hk0j01rlj(a)drn.newsguy.com>, stevendaryl3016(a)yahoo.com (Daryl McCullough) wrote: > mueckenh(a)rz.fh-augsburg.de says... > > >Virgil schrieb: > > > >> > Cantor said: A well-ordered set remains well-ordered, if finitely many > >> > or infinitely many transpositions are executed. > >> > >> Source??? > > > >If actual infinity does exist, then also an actually infinite set of > >transpositions must exist. Cantor knew that. > > The issue is whether "A well-ordered set remains well-ordered, if ... > infinitely many transpositions are executed." That's a provably false > statement, and I doubt that Cantor ever said it. Actually, even if it IS true, it would not allow transmutation of a set with a first element into one without a first element, at least if the transpositions are applied sequentially, as they must be. There would have to be a first transposition in the sequence which effects the change, and a single transposition is incapable of doing this.
From: Dann Corbit on 23 Jun 2006 19:44
"Richard Tobin" <richard(a)cogsci.ed.ac.uk> wrote in message news:e7hndv$4i3$2(a)pc-news.cogsci.ed.ac.uk... > In article <e7h9gb$pst$1(a)nntp.aioe.org>, Dann Corbit <dcorbit(a)connx.com> > wrote: >><mueckenh(a)rz.fh-augsburg.de> wrote in message >>news:1151079756.995887.177600(a)p79g2000cwp.googlegroups.com... >>[snip] >>> It is ridiculous to believe in Copernicus in the frame work of the >>> bible. Nevertheless I believe in Copernicus. > >>>What (exactly) is the place in the Bible that contradicts anything said >>>by >>Copernicus? > > Joshua 10:13 describes a miraculous event in which Joshua commands the > sun to stand still, with the result that the day continues until the > children of Israel had avenged themselves upon their enemies. This, > it was argued, shows that the cycle of day and night is normally > caused by the movement of the sun. It shows no such thing. The sun moves in the sky (visually). He was just asking for extended sunshine. |