Date: Feb 17, 2013 1:12 PM
Subject: Re: CON(ZF) and the ontology of ZF

On Feb 17, 11:42 am, fom <> wrote:
> So, returning to the statements in the opening
> paragraph, it does not surprise me that Zuhair
> may have succeeded in devising a means by which
> to show Con(ZF) relative to Morse-Kelley set theory.
> Morse-Kelley set theory as presented in Kelley
> presumes a global axiom of choice.

The theory that I've presented can actually work without the axiom
of global choice!

this is done by replacing axiom of Universal limitation by
axiom of direct size limitation.

To re-iterate my theory. It is too simple actually.

Language: FOL(=,e)

Definition: Set(x) <-> Ey(x e y)


1.Extensionality: (Az. z e x <-> z e y) -> x=y
2.Class comprehension: {x| Set(x) phi} exists.
3.Pairing: (Ay. y e x -> y=a or y=b) -> Set(x)
4.Hereditary limitation: Set(x) <-> Ey. Set(y) & AzeTC(x).z=<y
5.Size limitation: Set(x) & y=<x -> Set(y)

where x =< y <-> Ef. f:x-->y & f is injective
and TC(x)={y|As. x subset_of s & s is transitive -> y e s}

This proves MK-choice. However it might be stronger than MK-choice?
MK+global choice proves all the above axioms.