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).