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Topic: Just another exposition of MK.
Replies: 10   Last Post: Mar 17, 2013 1:50 PM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: Just another exposition of MK.
Posted: Mar 17, 2013 7:04 AM
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On Mar 17, 9:30 am, fom <fomJ...@nyms.net> wrote:
> On 3/16/2013 1:08 PM, Zuhair wrote:
>
>
>
>
>
>
>
>
>

> > On Mar 16, 9:33 am, Zuhair <zaljo...@gmail.com> wrote:
> >> Define: Set(x) iff {x,..}
>
> >> Extensionality: x C y & y C x -> x=y
>
> >> Comprehension: {x| Set(x) & phi}
>
> >> Pairing: x C {a,b} -> Set(x)
>
> >> Generation: Set(x) & y C H(x) -> Set(y)
>
> >> where H(x)={z| m in TC({z}). |m| =< |x|}
>
> >> Size: |x| < |V| -> Set(U(x))
>
> >> where TC, U stand for transitive closure, union respectively defined
> >> in the customary manner; C is subclass relation; | | =< | | and | | <
> >> | | relations are defined in the standard manner.

>
> >> The theory above minus axiom of Size is sufficient to prove
> >> consistency of Z. With the axiom of Size it can prove the consistency
> >> of ZF+Global choice, and it is equi-interpretable with MK+Global
> >> choice.

>
> >> Zuhair
>
> > Another reformulation along the same lines is:
>
> > Define: Set(X) iff {X,..} exists.
>
> > Extensionality: X C Y & Y C X -> X=Y
> > Class comprehension: {x|Set(x) phi} exists.
> > Pairing: X C {a,b} -> Set(X)
> > Subsets: Set(X) & Y C X -> Set(Y)
> > Size limitation: |X|<|V| -> Set(H(TC(X)))

>
> > C is sublcass relation.
> > TC(X) is the transitive closure of X.
> > H(X) is the Class of all sets hereditarily subnumerous to X.

>
> > Possibly (I'm not sure) pairing is redundant.
>
> How does your pairing axiom assert the
> existence of pairs?
>
> Are you assuming that the monadic
> Set() predicate is satisfied for
> every finite and transfinite
> enumeration?


I don't know what enumerations has got to do with pairing here. My
axiom is saying that any Class of two or less elements is a set, and
that's all what we need for pairing and building relations.

Zuhair



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