FOM: Finite axiom systems for sets
JoeShipman at aol.com
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).
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