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Topic: A reformulation of MK-Foundation-Choice: Even more compact!
Replies: 3   Last Post: Mar 28, 2013 2:25 PM

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Posts: 2,665
Registered: 6/29/07
Re: A reformulation of MK-Foundation-Choice: Even more compact!
Posted: Mar 28, 2013 7:07 AM
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On Mar 23, 8:33 pm, Zuhair <> wrote:
> This is even more compact reformulation of MK-Foundation-Choice.
> Unique Comprehension: if phi is a formula in which x is not free,
> then:
> (Exist x for all y (y in x iff set(y) & phi)) is an axiom.
> Size limitation: Set({}) & [Set(x) & y =< H(TC(x)) -> Set(y)]
> /

It might be possible to further weaken that to the following

Set({}) & [Set(x) & y=<H(x) -> Set(y)]

I think this can interpret MK over the sub-domain of well founded
sets, thus proving the consistency of ZFC relative to it.

Also I do think that if we re-define =< to the following modified
subset relation, then the resulting theory would prove the consistency
of ZC relative to it.

Def.) y =< x iff Exist z (for all m. m in y & ~m in x -> m=z)


> Def.) y C x iff for all z (z in y -> z in x)
> Def.) y =< x iff y C x Or Exist f (f:y-->x & f is injective)
> Def.) TC(x)= {y| for all t. t is transitive & x C t -> y in t}
> Def.) t is transitive iff for all m,n(m in n & n in t -> m in t)
> Def.) H(x)={y| for all z. z in TC(y) or z=y -> z =< x}
> It is nice to see that only one axiom can prove the existence of all
> sets in ZF.
> So Pairing, Union, Power, Infinity, Separation and Replacement All are
> provable over sets. Of course Foundation and Choice are interpretable
> in this theory.
> Con(ZFC) is actually Provable in this theory.
> Zuhair

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