```Date: Feb 19, 2013 3:53 AM
Author: fom
Subject: Re: CON(ZF) and the ontology of ZF

On 2/17/2013 12:53 PM, Zuhair wrote:> On Feb 17, 9:40 pm, fom <fomJ...@nyms.net> wrote:>> On 2/17/2013 12:12 PM, Zuhair wrote:>>>>> 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 axiom>>> of global choice!>>>> I believe this.  You represented the forcing methodology>> directly.  And, I am now fairly convinced that that>> methodology is implicit to the axiom of induction for>> arithmetic.>>>> Think carefully about how I ended that post.  I pointed>> to a link explaining the relationship of AC to GCH>>>> There is a reason I did that.  I do not ascribe>> to the usual model theory for set theory.  It is not>> logically secure.  Very few people like my posts,>> but this is one attempt at explaining myself on>> "truth" for set theory.>>>> news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdn...@giganews.com>>>>>>>>>>>>>>>>>>>>>>>>> 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)>>>>> Axioms:>>>>> 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.I think you need to make a change here, Zuhair.If you try to define that bounding set as the intersectionof all sets satisfying that property, you will collapse tosomething similar to my axiom,AxAy(xcy -> xe(GC(TC({y}))))would you not?(whoops! You did not see this because I did not put it inthe original posting.  I formulated an unreviewed, unpublishedpaper in 1994.  That had been my size limitation attempt.)Lets call your bounding set a Zuhair successor abbreviated to ZS.Then, in addition to what you already specify, you want somethinglike{x}e(GC(TC({y})))so that ZS(x) is independent of x relative to the transitiveclosure operation.  By independent, I mean always requiringsomething bigger than what x can generate through its singleton.Think of the set universe dynamically.  There is an involutionstreaming down from the choice function on the set of Zermelonames through the Zermelo names... {{{{x}}}} -> {{{x}}} -> {{x}} -> {x} ->It is at the singleton where the transitive closure operationstarts branching if x is not simple.I know that sounds funny, but, it is just the iterativeapplication of the inverse transformation for the transformationone would describe by{x} is the name of x{{x}} is the name of {x}{{{x}}} is the name of {{x}}{{{{x}}}} is the name of {{{x}}}where I am using braces instead of the quotes a logicianmight prefer.Your bounding set cannot be captured by TC({x}).
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