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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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 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
A reformulation of MK-Foundation-Choice: Even more compact!
Posted: Mar 23, 2013 1:33 PM

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)]
/

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

Date Subject Author
3/23/13 Zaljohar@gmail.com
3/23/13 Zaljohar@gmail.com
3/28/13 Zaljohar@gmail.com
3/28/13 Zaljohar@gmail.com