Stephen G. Simpson
 Department of Mathematics
 Penn State University

 Special Session on Reverse Mathematics
 Association for Symbolic Logic
 2001 Annual Meeting
 Philadelphia, PA
 March 10-13, 2001

 Title: An Overview of Reverse Mathematics


 Foundations of mathematics is the study of the most basic concepts
 and logical structure of mathematics as a whole.  Reverse mathematics
 is the foundational program of discovering which set existence axioms
 are needed to prove known theorems in core mathematical areas such as
 algebra, analysis, geometry, countable combinatorics.  It turns out
 that a large number of theorems fall into a small number of
 equivalence classes with respect to provable equivalence over a weak
 base theory.  (See my book Subsystems of Second Order Arithmetic,
 Springer-Verlag, 1999.)  The equivalence classes appear to reflect
 well known philosophical/foundational programs such as constructivism
 (Bishop), finitistic reductionism (Hilbert), predicativism (Weyl,
 Feferman), predicative reductionism, and impredicative analysis.  In
 this talk I comment on the mathematical, philosophical, and
 foundational significance of reverse mathematics as it has developed
 from the early 1970's to the present.  I also comment on the
 significance of reverse mathematics for the principal subdivisions of
 mathematical logic: model theory, set theory, recursion theory, proof