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Topic: A natural theory proving Con(ZFC)
Replies: 3   Last Post: Feb 11, 2013 12:25 PM

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Charlie-Boo

Posts: 1,588
Registered: 2/27/06
Re: A natural theory proving Con(ZFC)
Posted: Feb 11, 2013 12:25 PM
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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?

C-B

> 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





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