Date: Feb 8, 2013 9:33 AM
Subject: Re: A natural theory proving Con(ZFC)

On Feb 8, 4:27 pm, Frederick Williams <>
> Zuhair wrote:

> > I see the following theory a natural one that proves the consistency
> > of ZFC.

> > Language: FOL(=,in)
> How do you express Con(ZFC) in that language?  I know one can encode it
> using names of sets rather as one can encode Con(PA) using numerals, but
> isn't it rather hard work and is your claim justified without at least
> an outline?

The language of this theory is the same language of ZFC. Consistency
of ZFC would be proved by constructing a model of ZFC in this theory,
the class of all well founded sets in this theory is a model of ZF and
ZFC simply follows.

Note: if one desires a direct way to prove choice and global choice,
then the last axiom can be replaced by the following:

Universal limitation: x strictly < W -> set(x).

where "strictly <" refers < relation with absence of bijection, W is
the universal class of all sets.

Note: we can also use the ordinary relations defining subnumerousity
after injections instead of surjections used here and this would also
be enough to prove Con(ZFC).