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 > >  Extensionality: as in Z > >  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. > >  Pairing: (for all y. y in x -> y=a or y=b) -> Set(x) > >  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 would be 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] > > /