On Bar Recursion and Choice in a Classical Setting

Modified Bar Recursion and Classical Dependent Choice

Bar Recursion in Classical Realisability: Dependent Choice and Continuum Hypothesis

This paper is about the bar recursion operator in the context of classical realizability. The pioneering work of Berardi, Bezem, Coquand [1] was enhanced by Berger and Oliva [2]. Then Streicher [12] has shown, by means of their bar recursion operator, that the realizability models of ZF, obtained from usual models of λ-calculus (Scott domains, coherent spaces, . . . ), satisfy the axiom of depe...

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 ...

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...

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...