```Date: Mar 28, 2013 2:25 PM
Author: Zaljohar@gmail.com
Subject: Re: A reformulation of MK-Foundation-Choice: Even more compact!

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 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)]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 axiomSet({}) & [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 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)>> Zuhair>
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