
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 nonascii 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.
Nothing remotely circular about it.
> Impredicative definitions >allow "metatricks". > >(I know there are plenty other proofs...) > >LV >

