```Date: Jan 20, 2013 9:34 AM
Author: Zaljohar@gmail.com
Subject: Re: Hereditary size notion proving consistency of ZF

On Jan 19, 9:20 pm, Zuhair <zaljo...@gmail.com> wrote:> 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)>If we remove the condition y < x and replace Uy with x then this wouldbe sufficient to prove the consistency of 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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