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Topic: Objection to Cantor's Theorem
Replies: 62   Last Post: Jul 30, 2012 9:23 AM

 Messages: [ Previous | Next ]
 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Objection to Cantor's Theorem
Posted: Jul 18, 2012 10:45 AM

On Wed, 18 Jul 2012 03:03:39 +0100, "LudovicoVan"
<julio@diegidio.name> wrote:

><http://en.wikipedia.org/wiki/Cantor%27s_theorem>
>(Amended non-ascii characters.)
>
><< To establish Cantor's theorem it is enough to show that, for any given
>set A, no function f from A into P(A), the power set of A, can be
>surjective, i.e. to show the existence of at least one subset of A that is
>not an element of the image of A under f. Such a subset, B in P(A), is
>given by the following construction:
>
> B := { x in A | ~ x in f(x) }.
>
>This means, by definition, that for all x in A, x in B if and only if ~ x in
>f(x). >>
>
>So, the proof alleges to establish the there is no f from A into P(A) that
>can be surjective by postulating a set B that would not be in the image of
>f.

Huh? The existence of B is not "postulated". The set B is _defined_.

>This is invalid (circular) reasoning.

>-- Impredicative definitions
>allow "meta-tricks".
>
>(I know there are plenty other proofs...)
>
>-LV
>

Date Subject Author
7/17/12 LudovicoVan
7/17/12 LudovicoVan
7/17/12 Virgil
7/17/12 Virgil
7/17/12 LudovicoVan
7/17/12 LudovicoVan
7/18/12 Virgil
7/17/12 Virgil
7/17/12 LudovicoVan
7/18/12 Virgil
7/18/12 Michael Stemper
7/18/12 Virgil
7/18/12 dilettante
7/18/12 Virgil
7/19/12 Michael Stemper
7/18/12 Daryl McCullough
7/18/12 LudovicoVan
7/18/12 Shmuel (Seymour J.) Metz
7/18/12 Marshall
7/18/12 LudovicoVan
7/18/12 Shmuel (Seymour J.) Metz
7/18/12 David C. Ullrich
7/18/12 LudovicoVan
7/18/12 David C. Ullrich
7/18/12 LudovicoVan
7/18/12 David C. Ullrich
7/18/12 LudovicoVan
7/19/12 Virgil
7/19/12 W. Dale Hall
7/19/12 Gus Gassmann
7/19/12 Shmuel (Seymour J.) Metz
7/19/12 David C. Ullrich
7/20/12 LudovicoVan
7/20/12 David C. Ullrich
7/20/12 LudovicoVan
7/21/12 hagman
7/22/12 LudovicoVan
7/27/12 hagman
7/27/12 Virgil
7/27/12 LudovicoVan
7/27/12 Eddie
7/28/12 LudovicoVan
7/28/12 Eddie
7/28/12 Eddie
7/28/12 Virgil
7/28/12 LudovicoVan
7/28/12 Virgil
7/28/12 LudovicoVan
7/28/12 Virgil
7/28/12 David C. Ullrich
7/28/12 LudovicoVan
7/28/12 Virgil
7/29/12 hagman
7/29/12 LudovicoVan
7/29/12 hagman
7/30/12 LudovicoVan
7/22/12 Shmuel (Seymour J.) Metz
7/20/12 Shmuel (Seymour J.) Metz
7/18/12 Virgil
7/18/12 Virgil
7/27/12 Eddie
7/27/12 Eddie
7/27/12 Eddie