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Predicative Functionals and an Interpretation of ID<

Summary: Predicative Functionals and
an Interpretation of ID<

Jeremy Avigad
December 22, 1997
In 1958 GĻodel published his Dialectica interpretation, which reduces
classical arithmetic to a quantifier-free theory T axiomatizing the prim-
itive recursive functionals of finite type. Here we extend GĻodel's T to
theories Pn of "predicative" functionals, which are defined using Martin-
LĻof's universes of transfinite types. We then extend GĻodel's interpretation
to the theories of arithmetic inductive definitions IDn, so that each IDn
is interpreted in the corresponding Pn. Since the strengths of the theories
IDn are cofinal in the ordinal 0, as a corollary this analysis provides an
ordinal-free characterization of the <0-recursive functions.
1 Introduction
1.1 Background
In 1958, GĻodel [18] published what is now known as the Dialectica interpretation
of arithmetic, consisting of a quantifier-free theory T and interpretation of Peano
Arithmetic (PA) in that theory. T allows for the definition of functionals of


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


Collections: Multidisciplinary Databases and Resources; Mathematics