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Hereditary size notion proving consistency of ZF
Posted:
Jan 19, 2013 1:20 PM


I think that ZF can be proved consistent relative to consistency of the following theory defined in the same language of ZF with the following axioms:
Define: Set(x) iff Exist y. x in y
[1] Extensionality: as in Z
[2] 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.
[3] Pairing: (for all y. y in x > y=a or y=b) > Set(x)
[4] Size: [Set(x) & y < x & for all m in z (m << Uy)] > Set(z)
Definitions:
y < x is defined as: Exist s,f: s subset_of x & f:s > y & f is surjective.
y << x is defined as: y < x & (for all z in TC(y). z < x)
TC(y)=x iff [for all z. z in x iff (for all s. y subset_of s & s is transitive > z in s)]
Uy=x iff [for all z. z in x iff Exist k. k in y & z in k]
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