 
Summary: A ModelTheoretic Approach to
Ordinal Analysis
Jeremy Avigad and Richard Sommer
February 4, 1997
Abstract
We describe a modeltheoretic approach to ordinal analysis via the
finite combinatorial notion of an large set of natural numbers. In con
trast to syntactic approaches that use cut elimination, this approach
involves constructing finite sets of numbers with combinatorial prop
erties that, in nonstandard instances, give rise to models of the theory
being analyzed. This method is applied to obtain ordinal analyses of a
number of interesting subsystems of first and secondorder arithmetic.
1 Introduction
Two of proof theory's defining goals are the justification of classical
theories on constructive grounds, and the extraction of constructive
information from classical proofs. Since Gentzen, ordinal analysis has
been a major component in these pursuits, and the assignment of re
cursive ordinals to theories has proven to be an illuminating way of
measuring their constructive strength. The traditional approach to
ordinal analysis, which uses cutelimination procedures to transform
