Abstract

AbstractExtending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

Highlights

  • Let X be a set, and let Γ be a monotone operator from the power set of X to itself, so that A ⊆ B implies Γ(A) ⊆ Γ(B)

  • I can be characterized as the limit of a sequence indexed by a sufficiently long segment of the ordinals, defined by I0 = ∅, Iα+1 = Γ(Iα), and Iλ = γ

  • The operator Γ is given by a positive arithmetic formula φ(x, P ), in the sense that Γ(A) = {x | φ(x, A)} and φ is an arithmetic formula in which the predicate P occurs only positively. (The positivity requirement can be expressed by saying that no occurrence of P is negated when φ is written in negation-normal form.)

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Summary

Introduction

Let X be a set, and let Γ be a monotone operator from the power set of X to itself, so that A ⊆ B implies Γ(A) ⊆ Γ(B). We present a new method of carrying out this first step, based on a functional interpretation along the lines of Godel’s “Dialectica” interpretation of firstorder arithmetic Such functional interpretations have proved remarkably effective in “unwinding” computational and otherwise explicit information from classical arguments (see, for example, [22, 23, 24]). Feferman [12] used a Dialectica interpretation to obtain ordinal bounds on the strength of ID1 (the details are sketched in [6, Section 9]), and Zucker [36] used a similar interpretation to bound the ordinal strength of ID2 These interpretations do not yield Π2 reductions to constructive theories, and do not provide computational information; nor do the methods seem to extend to the theories beyond ID2.

Background
Embedding ID1 in OID1
A functional interpretation of OID1
Iterating the interpretation

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