Related papers: A Guide to Krivine Realizability for Set Theory
J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new…
Realizability, introduced by Kleene, can be understood as a concretization of the Brouwer-Heyting-Kolmogorov (BHK) interpretation of proofs, providing a framework to interpret mathematical statements and proofs in terms of their…
Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene's initial realizability laid the ground…
Introduced by Dal Lago and Hofmann, quantitative realizability is a technique used to define models for logics based on Multiplicative Linear Logic. A particularity is that functions are interpreted as bounded time computable functions. It…
We define an ordinalized version of Kleene's realizability interpretation of intuitionistic logic by replacing Turing machines with Koepke's ordinal Turing machines (OTMs), thus obtaining a notion of realizability applying to arbitrary…
We introduce a notion of realizability with ordinal Turing machines based on recognizability rather than computability, i.e., the ability to uniquely identify an object. We show that the arising concept of $r$-realizabilty has the property…
We investigate a framework of Krivine realizability with I/O effects, and present a method of associating realizability models to specifications on the I/O behavior of processes, by using adequate interpretations of the central concepts of…
Besides recalling the basic definitions of Realizability Lattices, Abstract Krivine Structures, Ordered Combinatory Algebras and Tripos and reviewing its relationships, we propose a new foundational framework for realizability. Motivated by…
This paper presents a formal theory of Krivine's classical realisability interpretation for first-order Peano arithmetic ($\mathsf{PA}$). To formulate the theory as an extension of $\mathsf{PA}$, we first modify Krivine's original…
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.
The technique of "classical realizability" is an extension of the method of "forcing"; it permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory and to build new models of ZF, called…
The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the…
In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability…
We present tools for analysing ordinals in realizability models of classical set theory built using Krivine's technique for realizability. This method uses a conservative extension of $ZF$ known as $ZF_{\varepsilon}$, where two membership…
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…
In this paper we continue with the algebraic study of Krivine's realizability, refining some of the authors' previous constructions by introducing two categories, with objects the abstract Krivine structures and the implicative algebras…
This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability…
We prove the following completeness result about classical realizability: given any Boolean algebra with at least two elements, there exists a Krivine-style classical realizability model whose characteristic Boolean algebra is elementarily…
For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness.…
A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While…