FOM: Infinity

[JoeShipman@aol.com]

Heck: 
>>So I take it that the sensible finite axiomatization of GB to which Shipman alludes below isn't the one I know, since that isn't any more finite than the first-blush axiomatization of ACA_0. What is it, then?<<

This is done on pp.74-75 of Cohen's book "Set Theory and the Continuum Hypothesis".  

To the language with the binary membership relation, you add a unary sethood relation M(x)and an axiom saying only sets can be members of something.  You then add the usual ZFC axioms for sets, omitting replacement.  

Instead of Replacement, you have an axiom saying that if a class X of ordered pairs represents a function on all sets (that is, every set is the first component of exactly one of the members of the class) then for every set u the range of the function X on u exists and is a set.

Finally, you add 8 axioms for class formation:

1) There is a class representing the membership relation on sets
2) The intersection of two classes is a class
3) The complement of a class is a class
4) Given a class X, there is another class containing exactly the second components of the elements of X which are ordered pairs
5) Given a class X, there is another class containing all ordered pairs whose second component is in X.
6) Given a class X, there is another class containing exactly those ordered pairs <b,c> where the ordered pair <c,b> is in X.
7) Given a class X, there is another class containing exactly those ordered triples <b,c,a> where the ordered triple <a,b,c> is in X.
8) Given a class X, there is another class containing exactly those ordered triples <a,c,b> where the ordered pair <a,b,c> is in X.

- Joe Shipman