On Feb 8, 9:33 am, Zuhair <zaljo...@gmail.com> wrote: > 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.
Who will bell the cat?
> 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