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Re: A reformulation of MKFoundationChoice: Even more compact!
Posted:
Mar 28, 2013 2:25 PM


On Mar 28, 2:07 pm, Zuhair <zaljo...@gmail.com> wrote: > 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)]
No this won't work we need H(TC(x)) as in the original formulation.
But for the sake of proving Con(ZC) yes we can use the weak axiom
Set({}) & [Set(x) & y =< H(x) > Set(y)]
where =< is defined as:
y =< x iff Exist z (for all m. m in y & ~m in x > m=z)
Zuhair > > 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 >



