
Re: Just another exposition of MK.
Posted:
Mar 17, 2013 7:04 AM


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 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. > > 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

