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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

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

Date Subject Author
3/16/13 Zaljohar@gmail.com
3/16/13 fom
3/16/13 Zaljohar@gmail.com
3/16/13 Zaljohar@gmail.com
3/17/13 ross.finlayson@gmail.com
3/16/13 Zaljohar@gmail.com
3/17/13 fom
3/17/13 Zaljohar@gmail.com
3/17/13 fom
3/17/13 Frederick Williams
3/17/13 Zaljohar@gmail.com