Date: Feb 8, 2013 9:33 AM
Author: Zaljohar@gmail.com
Subject: Re: A natural theory proving Con(ZFC)
On Feb 8, 4:27 pm, Frederick Williams <freddywilli...@btinternet.com>

wrote:

> 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).

Zuhair