Set Theory and Arithmetic
by
M. Randall Holmes
Boise State University

Ali Enayat and Robert Solovay have recently announced some results about equiconsistency of systems of arithmetic with versions of the Quine-Jense set theory NFU with the negation of the axiom of infinity. I have been able to verify some of the claimed results on my own and will offer some discussion. For example, NFU + "the universe is finite" + "every cantorian natural number is strongly cantorian" is precisely equiconsistent with PA. An interesting aspect of this is that the NFU theory, unlike PA, is finitely axiomatizable. A possible application of these systems is as an alternative foundation for "nonstandard analysis".

Date received: March 21, 2002