Stephen G. Simpson: The Gödel Hierarchy and Reverse Mathematics.


Hilbert's concern for consistency proofs led to Gödel's Second Incompleteness Theorem, which led to the study of what we may now call the Gödel Hierarchy. The Gödel Hierarchy is a structure consisting of foundational theories in the first-order predicate calculus, ordered by consistency strength or relative interpretability. The Gödel Hierarchy appears to be linear and to exhibit other remarkable regularities.

Reverse Mathematics is a modern program of foundational research which seeks to classify standard, core mathematical theorems into equivalence classes via logical equivalence modulo a weak base theory. Surprisingly, a small number of classes occur again and again. Furthermore, these frequently occurring classes correspond to natural stopping points in the Gödel Hierarchy, which in turn correspond to venerable foundational programs, including Hilbert's Program of finitistic reductionism, and Weyl's Program of predicativity. As a byproduct of Reverse Mathematics, one has discovered an extensive partial realization of Hilbert's Program.