Arithmetical conservation results

In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection of provable arithmetical sentences. Our proof relies on several ideas from earlier proofs by other authors, but adds some new ones as well. In particular, we show how a recent paper by Jaap van Oosten can be used to simplify a key step in the proof. We have also included an interesting corollary for classical systems pointed out to us by Ulrich Kohlenbach

Bibliography on Realizability

AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in “A Workshop on Realizability Semantics and Its Applications”, Trento, Italy, June 30–July 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html

Extensional realizability

AbstractTwo straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is (as a relation between numbers and sentences) a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability

Extended Bar Induction in Applicative Theories

TAPP is a total applicative theory, conservative over intuitionistic arithmetic. In this paper, we first show that the same holds for TAPP + the choice principle EAC; then we extend TAPP with choice sequences and study the principle EBI0a (arithmetical extended bar induction of type zero). The resulting theories are used to characterise the arithmetical fragment of EL (elementary intuitionistic analysis) + EBI0a. As a digression, we use TAPP to show that P. Martin-Löf’s basic extensional theory ML0 is conservative over intuitionistic arithmetic

Extended Bar Induction in Applicative Theories

TAPP is a total applicative theory, conservative over intuitionistic arithmetic. In this paper, we first show that the same holds for TAPP + the choice principle EAC; then we extend TAPP with choice sequences and study the principle EBI0a (arithmetical extended bar induction of type zero). The resulting theories are used to characterise the arithmetical fragment of EL (elementary intuitionistic analysis) + EBI0a. As a digression, we use TAPP to show that P. Martin-Löf’s basic extensional theory ML0 is conservative over intuitionistic arithmetic.