We will discuss an approach to proof-theoretic analysis which identifies the search for proof-theoretic ordinals with the study of axioms of infinity. More specifically, we are concerned with axioms of infinity which are stated in terms of embeddings and patterns formed by finite collections of embeddings. The simplest case is provided by a continuous hierarchy of structures Mi (i an ordinal) where the a embeddings are given by the pairs (i,j) such that Mi is a Sigma-1 elementary substructure of Mj. Let <1 be the collection of such pairs. When ... is a list of reasonable operations on the class of ordinals, ORD, the order in which isomorphism types of finite substructures of (ORD,...,<,<1) arise is independent of the hierarchy under some fairly mild assumptions. These patterns provide descriptions of important proof-theoretic ordinals e.g. when ... is empty the resulting ordinal is epsilon_0 and when ... consists of 0 and + the result is the proof-theoretic ordinal of KPl_0 (or, equivalently, (Pi-1-1)-CA_0). These patterns are called "patterns of resemblance of order 1". We will discuss patterns of resemblance of arbitrary finite orders which reflect when a given element of a hierarchy is a Sigma-n elementary substructure of another for arbitrary n, a program to show these patterns provide the proof-theoretic ordinal of formal second-order arithmetic, and extensions to include more complex embeddings.

Back to Talks Page