FOM: Finite axiom systems for sets JoeShipman at
Tue May 22 14:25:40 EDT 2001

Are there interesting finitely axiomatizable theories of sets that allow one to do all or most of "ordinary mathematics"?

Yes - lots of extensions of NFU for example.

This isn't what I was looking for.  A "set" in NFU isn't the same thing as a set in ZFC.  If I was willing to change the meaning of "set" I would just use VNBG which finitely axiomatizes classes and in which ordinary sets are definable as those classes which are elements of something.

What I want is a finite set of sentences which are supposed to be true in the ordinary set-theoretical universe V, which axiomatize a natural theory strong enough to do ordinary mathematics.  I suggested two possibilities, neither of which is terribly "natural", and asked if they had nicer equivalents:

1) ZC'  -- the non-schematized axioms of ZFC, plus enough instances of replacement to define truth in V_(omega+omega) and prove that V_(omega+omega) models ZC -- this gets all the theorems of ZC that only speak about small sets

2) ZFn, where replacement is restricted to pi_n formulas, for a sufficiently large n.

Here's another idea.  How about using INAC="there exists an inaccessible cardinal"?  From INAC, ZFC-Replacement,  and finitely many instances of the Replacement scheme (just enough to prove that the sets of accessible rank form a model of ZFC) I can derive any theorem of ZFC that speaks only about sets of accessible rank (rank less than the first inaccessible cardinal).

-- JS

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