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Krivine's classical realisability from a categorical perspective

Published online by Cambridge University Press:  10 April 2013

THOMAS STREICHER
THOMAS STREICHER*
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany Email: streicher@mathematik.tu-darmstadt.de
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Abstract

In a sequence of papers (Krivine 2001; Krivine 2003; Krivine 2009), J.-L. Krivine introduced his notion of classical realisability 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 realisability interpretation for the axiom of choice.

The aim of the current paper is to show how Krivine's classical realisability can be understood as an instance of the categorical approach to realisability as started by Martin Hyland in Hyland (1982) and described in detail in van Oosten (2008). Moreover, we will give an intuitive explanation of the iteration of realisability as described in Krivine (2008).

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Issue 6 , December 2013 , pp. 1234 - 1256
Copyright
Copyright © Cambridge University Press 2013 

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References

van den Berg, B. and Moerdijk, I. (2009) A unified approach to algebraic set theory. In: Cooper, S. B., Geuvers, H., Pillay, A. and Väänänen, J. (eds.) Logic Colloquium 2006, Lecture Notes in Logic 32, Cambridge University Press 18–37.CrossRefGoogle Scholar
Griffin, T. (1990) A formulae-as-types notion of control. In: POPL ‘90: Proceedings of the 17th ACM SIGPLAN–SIGACT symposium on Principles of Programming Languages, ACM 47–58.Google Scholar
Hofstra, P. and van Oosten, J. (2003) Ordered partial combinatory algebras. Mathematical Proceedings of the Cambridge Philosophical Society 134 (3)445–463.CrossRefGoogle Scholar
Hofstra, P. (2006) All realizability is relative. Mathematical Proceedings of the Cambridge Philosophical Society 141 (2)239–264.CrossRefGoogle Scholar
Hofstra, P. (2008) Iterated realizability as a comma construction. Mathematical Proceedings of the Cambridge Philosophical Society 144 (1)39–51.CrossRefGoogle Scholar
Hyland, J. M. E. (1982) The effective topos. In: Troelstra, A. S. and van Dalen, D. (eds.) Proceedings of the The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout 1981), Studies in Logic and the Foundations of Mathematics 110, North-Holland 165–216.Google Scholar
Krivine, J.-L. and Parigot, M. (1990) Programming with proofs. Journal of Information Processing and Cybernetics/Elektronische Informationsverarbeitung und Kybernetik 26 (3)149–167.Google Scholar
Krivine, J.-L. (1990a) Lambda-calcul, types et modèles, Masson.Google Scholar
Krivine, J.-L. (1990b) Opérateurs de mise en mémoire et traduction de Gödel. Archive for Mathematical Logic 30 (4)241–267.CrossRefGoogle Scholar
Krivine, J.-L. (2001) Types lambda-calculus in classical Zermelo–Fraenkel set theory. Archive for Mathematical Logic 40 (3)189–205.CrossRefGoogle Scholar
Krivine, J.-L. (2003) Dependent choice, ‘quote’ and the clock. Theoretical Computer Science 308 259–276.CrossRefGoogle Scholar
Krivine, J.-L. (2008) Structures de réalisabilité, RAM et ultrafiltre sur N. Avaialable from http://www.pps.jussieu.fr/~krivine/Ultrafiltre.pdf.Google Scholar
Krivine, J.-L. (2009) Realizability in classical logic. In: Interactive models of computation and program behaviour, Panoramas et synthèses 27, SMF.Google Scholar
Krivine, J.-L. (2011) Realizability algebras : a program to well order ℝ. Logical Methods in Computer Science 7 1–47.Google Scholar
Krivine, J.-L. (2010) Realizability algebras II: new models for ZF + DC. Avaialable from http://www.pps.jussieu.fr/~krivine/articles/R_ZF.pdf.Google Scholar
van Oosten, J. (2008) Realizability: An Introduction to its Categorical Side, Elsevier.Google Scholar
Streicher, T. and Reus, B. (1998) Classical logic, continuation semantics and abstract machines. Journal of Functional Programming 8 (6)543–572.CrossRefGoogle Scholar