A Classical Realizability Model arising from a Stable Model of Untyped Lambda Calculus

We study a classical realizability model (in the sense of J.-L. Krivine) arising from a model of untyped lambda calculus in coherence spaces. We show that this model validates countable choice using bar recursion and bar induction

Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

CZF does not have the Existence Property

Constructive theories usually have interesting metamathematical properties where explicit witnesses can be extracted from proofs of existential sentences. For relational theories, probably the most natural of these is the existence property, EP, sometimes referred to as the set existence property. This states that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since the 80's that EP holds for some intuitionistic set theories and yet fails for IZF. Despite this, it has remained open until now whether EP holds for the most well known constructive set theory, CZF. In this paper we show that EP fails for CZF

Existential witness extraction in classical realizability and via a negative translation

We show how to extract existential witnesses from classical proofs using Krivine's classical realizability---where classical proofs are interpreted as lambda-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Sigma01-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS), 201