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Topic: CON(ZF) and the ontology of ZF
Replies: 5   Last Post: Feb 19, 2013 3:53 AM

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Posts: 1,968
Registered: 12/4/12
Re: CON(ZF) and the ontology of ZF
Posted: Feb 17, 2013 1:40 PM
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On 2/17/2013 12:12 PM, Zuhair wrote:
> 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!

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

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.


> 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.

In general, I am unfamiliar with Morse-Kelley. I have
read through the appendix of "General Topology", and
that does have only global choice. So, I am not certain
of your distinctions here and cannot even begin to
address the question.

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