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 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]
>
> /