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Re: A reformulation of MKFoundationChoice: Even more compact!
Posted:
Mar 28, 2013 7:07 AM


On Mar 23, 8:33 pm, Zuhair <zaljo...@gmail.com> wrote: > This is even more compact reformulation of MKFoundationChoice. > > 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 subdomain of well founded sets, thus proving the consistency of ZFC relative to it.
Also I do think that if we redefine =< 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)
Zuhair
> 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



