In a sequence of papers (Krivine 2001; Krivine 2003; Krivine 2009) J.-L. Krivine has introduced his notion of Classical Realizability for classical second order logic and Zermelo-Fraenkel set theory. Moreover, in more recent work (Krivine 2008) he has considered forcing constructions on top of it with the ultimate aim of providing a realizability interpretation for the axiom of choice. The aim ...

This article systematically investigates so-called “truth variants” of several functional interpretations. We start by showing a close relation between two variants of modified realizability, namely modified realizability with truth and q-modified realizability. Both variants are shown to be derived from a single “functional interpretation with truth” of intuitionistic linear logic. This analys...

Models of a generalized nondeterminism are defined by limitations on nondeterministic behavior of a computing device. A regular realizability problem is a problem of verifying existence of a special sort word in a regular language. These notions are closely connected. In this paper we consider regular realizability problems for languages consisting of all prefixes of an infinite word. These pro...

We give a new presentation of interactive realizability with a more explicit syntax. Interactive realizability is a realizability semantics that extends the Curry-Howard correspondence to (sub-)classical logic, more precisely to first-order intuitionistic arithmetic (Heyting Arithmetic) extended by the law of the excluded middle restricted to simply existential formulas, a system motivated by i...

The notion of realizability algebra, which was introduced in [17, 18], is a tool to study the proof-program correspondence and to build new models of set theory, which we call realizability models of ZF. It is a variant of the well known notion of combinatory algebra, with a new instruction cc, and a new type for the environments. The sets of forcing conditions, in common use in set theory, are...

This paper investigates the question of whether Krivine’s classical realizability can provide a verificationist interpretation of classical logic. We argue that this kind of realizability can be considered an adequate candidate for this semantic role, provided that the notion of verification involved is no longer based on proofs, but on programs. On this basis, we show that a special reading of...

Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object’ functor from the topos of sets preserves finite coproducts, and that De Morgan’s law is satisfied. In this paper we show that these...