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A Model-Theoretic Approach to Ordinal Analysis

Summary: A Model-Theoretic Approach to
Ordinal Analysis
Jeremy Avigad and Richard Sommer
February 4, 1997
We describe a model-theoretic 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 second-order 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 cut-elimination procedures to transform


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics