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]

>

> /