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

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 Virgil Posts: 4,491 Registered: 1/6/11
Re: Objection to Cantor's Theorem
Posted: Jul 17, 2012 10:47 PM

In article <ju56i0\$7bm\$1@speranza.aioe.org>,
"LudovicoVan" <julio@diegidio.name> wrote:

> << 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.

It is certainly trivially true for finite sets, since for any function
of finite domain of size n, the power set is of size 2^n > n.

And given any function f: A -> P(A)
will the set B := { x in A | ~ x in f(x) }
(1) be well-defined ?
(2) in the mage of f ?
--

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