Re: Resolving the paradoxes of set theory

From: Nathan (ntspam2_at_netscape.net)
Date: 11/01/04 

Date: 1 Nov 2004 08:59:12 -0800

daryl@atc-nycorp.com (Daryl McCullough) wrote in message news:<cm32ud0pce@drn.newsguy.com>...
> George Greene says...
> >
> >Norman Megill <nm@nospam.see.signature.domain.invalid> wrote
>
> >> The proof that there is no surjection from a set to its power
> >> class does not require the power set axiom.
> >
> >Of course it does.
>
> No, it doesn't. The use of P(x) is just a notational convenience.
> State the theorem this way:
>
> forall x, forall f, if f is a function whose domain is x,
> then there exists a set x' such that x' is a subset of x
> and x' is not in the range of f
>
> The proof uses separation (given a set x and a formula Phi(y),
> there is a set x' consisting of all y in x satisfying Phi(y))
> but it doesn't use the power set axiom.

This is very nice, but without the power set axiom, what does it
really tell you? Cantor's Theorem is the justification of the
existence of unequal infinite cardinalities. If there is no set
of all subsets of x, then what application can this theorem have?

Relevant Pages

  • Re: Resolving the paradoxes of set theory
    ... George Greene says... ... >> class does not require the power set axiom. ... The use of Pis just a notational convenience. ... forall x, forall f, if f is a function whose domain is x, ...
    (sci.logic)
  • Re: Resolving the paradoxes of set theory
    ... What I mean by "notational convenience" is that any closed formula ... I'm not sure what you mean by saying that the power set axiom ... If f and x are sets, then so is x' by separation. ... >This is NOT going to automatically entail existence of any terms. ...
    (sci.logic)
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