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Re: CON(ZF) and the ontology of ZF
Posted:
Feb 17, 2013 1:12 PM


On Feb 17, 11:42 am, fom <fomJ...@nyms.net> wrote: > > So, returning to the statements in the opening > paragraph, it does not surprise me that Zuhair > may have succeeded in devising a means by which > to show Con(ZF) relative to MorseKelley set theory. > MorseKelley set theory as presented in Kelley > presumes a global axiom of choice.
The theory that I've presented can actually work without the axiom of global choice!
this is done by replacing axiom of Universal limitation by axiom of direct size limitation.
To reiterate my theory. It is too simple actually.
Language: FOL(=,e)
Definition: Set(x) <> Ey(x e y)
Axioms:
1.Extensionality: (Az. z e x <> z e y) > x=y 2.Class comprehension: {x Set(x) phi} exists. 3.Pairing: (Ay. y e x > y=a or y=b) > Set(x) 4.Hereditary limitation: Set(x) <> Ey. Set(y) & AzeTC(x).z=<y 5.Size limitation: Set(x) & y=<x > Set(y) /
where x =< y <> Ef. f:x>y & f is injective and TC(x)={yAs. x subset_of s & s is transitive > y e s}
This proves MKchoice. However it might be stronger than MKchoice? MK+global choice proves all the above axioms.
Zuhair



