fom
Posts:
1,969
Registered:
12/4/12


Re: Just another exposition of MK.
Posted:
Mar 17, 2013 8:16 AM


On 3/17/2013 6:04 AM, Zuhair wrote: > 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.
Typically, the axiom of pairing asserts the existence of pairs.
AxAyEzAw(wez <> (x=w \/ y=w))
Since your axiom is asserting exactly what you have characterized it as asserting, it seems the usual existence assertion must be arising elsewhere. Since I was uncertain about the exact meaning of the enumeration predicate, I thought it might have to do with that.

