Date: Jan 19, 2013 1:20 PM
Author: Zaljohar@gmail.com
Subject: Hereditary size notion proving consistency of ZF

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]

/