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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 16, 2013 2:08 PM
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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.

Zuhair



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