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A reformulation of MKFoundationChoice: Even more compact!
Posted:
Mar 23, 2013 1:33 PM


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)] /
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



