Bar Recursion in Classical Realisability: Dependent Choice and Continuum Hypothesis

Bar recursion in classical realisability : dependent choice and well ordering of R

This paper is about the bar recursion operator [9], in the context of classical realizability [6, 7]. It is a sequel to the three papers [1, 2, 10]. We use the definitions and notations of the theory of classical realizability as expounded in [5, 6, 7].1 In [1], S. Berardi, M. Bezem and T. Coquand have shown that a form of the bar recursion operator can be used, in a proof-program correspondenc...

Modified Bar Recursion and Classical Dependent Choice

Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice

During the last twenty years or so a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with some form of bar recursion, which realizes the negative translation of either countable or countable dependent choice. In this work we present the many varia...

On Bar Recursion and Choice in a Classical Setting

We show how Modified Bar-Recursion, a variant of Spector’sBar-Recursion due to Berger and Oliva can be used to realize the Axiomof Choice in Parigot’s Lambda-Mu-calculus, a direct-style language forthe representation and evaluation of classical proofs.We rely on Hyland-Ong innocent games. They provide a model to per-form the usual infinitary reasoning on Bar-Recursion needed...

Modified bar recursion

We introduce a variant of Spector’s bar recursion (called “modified bar recursion”) in finite types to give a realizability interpretation of the classical axiom of countable choice allowing for the extraction of witnesses from proofs of Σ1 formulas in classical analysis. As a second application of modified bar recursion we present a bar recursive definition of the fan functional. Moreover, we ...