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ORDINAL ANALYSIS WITHOUT PROOFS JEREMY AVIGAD
 

Summary: ORDINAL ANALYSIS WITHOUT PROOFS
JEREMY AVIGAD
Abstract. An approach to ordinal analysis is presented which is finitary, but high-
lights the semantic content of the theories under consideration, rather than the syntactic
structure of their proofs. In this paper the methods are applied to the analysis of the-
ories extending Peano arithmetic with transfinite induction and transfinite arithmetic
hierarchies.
1. Introduction. As the name implies, in the field of proof theory one
tends to focus on proofs. Nowhere is this emphasis more evident than in
the field of ordinal analysis, where one typically designs procedures for "un-
winding" derivations in appropriate deductive systems. One might wonder,
however, if this emphasis is really necessary; after all, the results of an ordinal
analysis describe a relationship between a system of ordinal notations and a
theory, and it is natural to think of the latter as the set of semantic conse-
quences of some axioms. From this point of view, it may seem disappointing
that we have to choose a specific deductive system before we can begin the
ordinal analysis.
In fact, Hilbert's epsilon substitution method, historically the first attempt
at finding a finitary consistency proof for arithmetic, has a more semantic char-
acter. With this method one uses so-called epsilon terms to reduce arithmetic

  

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

 

Collections: Multidisciplinary Databases and Resources; Mathematics