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The Model-Theoretic Ordinal Analysis of Theories of Predicative Strength

Summary: The Model-Theoretic Ordinal Analysis of
Theories of Predicative Strength
Jeremy Avigad and Richard Sommer
October 14, 1997
We use model-theoretic methods described in [3] to obtain ordinal
analyses of a number of theories of first- and second-order arithmetic,
whose proof-theoretic ordinals are less than or equal to 0.
1 Introduction
In [3] we introduced a model-theoretic approach to ordinal analysis
as an interesting alternative to cut elimination. Here we extend these
methods to the analysis of stronger theories of first- and second-order
arithmetic which are nonetheless predicatively justifiable.
When used in this sense, the word "predicative" refers to a foun-
dational stance under which one is willing to accept the set of natural
numbers as a completed totality, but not the set of all subsets of the
natural numbers. In this spirit, predicative theories bar definitions that
require quantification over the full power set of N, depicting instead
a universe of sets of numbers that is constructed "from the bottom
up." Work of Feferman and Sch¨utte has established that the ordinal


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


Collections: Multidisciplinary Databases and Resources; Mathematics