```Date: Feb 17, 2013 1:12 PM
Author: Zaljohar@gmail.com
Subject: Re: CON(ZF) and the ontology of ZF

On Feb 17, 11:42 am, fom <fomJ...@nyms.net> 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 axiomof global choice!this is done by replacing axiom of Universal limitation byaxiom of direct size limitation.To re-iterate my theory. It is too simple actually.Language: FOL(=,e)Definition: Set(x) <-> Ey(x e y)Axioms:1.Extensionality: (Az. z e x <-> z e y) -> x=y2.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=<y5.Size limitation: Set(x) & y=<x  -> Set(y)/where x =< y <-> Ef. f:x-->y & f is injectiveand 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.Zuhair
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