Re: The real numbers, and general comments

From: Andrew Usher (k_over_hbarc_at_yahoo.com)
Date: 10/05/04 

Date: 4 Oct 2004 21:04:58 -0700

Dave Seaman <dseaman@no.such.host> wrote in message news:<cjqcb5$2uo$1@mozo.cc.purdue.edu>...

> > The trouble is that you are convinced that ZFC is a good moel of
> > logic.
>
> I don't consider ZFC to be a model of anything. I was talking about ZF
> as a set of axioms that serves as a basis for set theory.

I know, you are a formalist, which I find objectionable.

> > Cantor's power-set theorem can be proved using real logic, as
> > Cantor did; it can also be 'proved' in ZF.
>
> It's an axiom of ZF.

No, a theorem. The PS axiom doesn't assert P(A) > A.

> > But (I think) Lowenheim-Skolen says that a bijection does exist, but
> > ZFC can't find it. Now if L-S says only that 'there exist countable
> > models', no problem; but I amd not sure which.
>
> That's not what L-S says. L-S is about models, and Cantor's theorem does
> not make reference to any particular model.

The version of Cantor not using a model is therefore not ZF.

Andrew Usher

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