
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 equiinterpretable 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: {xSet(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

