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