SPEAKER: Stephen G. Simpson (Pennsylvania State University)

 TITLE: The G\"odel Hierarchy and Reverse Mathematics

 ABSTRACT: The G\"odel Hierarchy is an array of foundationally
 significant theories in the predicate calculus.  The theories range
 from weak (bounded arithmetic, elementary function arithmetic)
 through intermediate (subsystems of second-order arithmetic), through
 strong (Zermelo/Fraenkel set theory, large cardinals).  The theories
 are ordered by inclusion, interpretability, and consistency strength.
 Reverse Mathematics is a program which seeks to classify mathematical
 theorems by calibrating their places within the G\"odel Hierarchy.
 The theorems are drawn from core mathematical areas such as analysis,
 algebra, functional analysis, topology, and combinatorics.
 Remarkably, the Reverse Mathematics classification scheme exhibits a
 considerable amount of regularity and structure.  In particular, a
 large number of core mathematical theorems fall into a small number
 of foundationally significant equivalence classes (the so-called
 ``big five'').  There are close connections with other foundational
 programs and hierarchies.  In particular, concepts and methods from
 degrees of unsolvability play an important role.