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**1 - 6**of**6**### Partial Combinatory Algebras and Realizability Toposes

, 2004

"... These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for ..."

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These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for some of the examples perhaps), although familiarity with combinatory logic and lambda calculus will not hurt. The second part on realizability toposes presupposes some knowledge of category theory; more specifically, we will assume that the reader knows what a topos is. Apart from that the material is self-contained. 1 Partial Combinatory Algebras We give the basic definitions and properties of Partial Combinatory Algebras in the first subsection. Next, we discuss some of the important examples. Finally, we touch upon the theory of Partial Combinatory Algebras. 1.1 Partial Applicative Structures and Combinatory Completeness We first introduce the basic concept of a Partial Applicative Structure, which may be viewed as a universe for computation. Then look at terms over an applicative structure, we formulate

### for Constructive Set Theory

, 2008

"... This article presents a generalisation of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalisation of the Boolean models for classical set theory which are a kind of forcing, while realizability is a decidedly constructive method that has first ..."

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This article presents a generalisation of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalisation of the Boolean models for classical set theory which are a kind of forcing, while realizability is a decidedly constructive method that has first been develloped for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalisation, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. The generalisation not only deepens the understanding of class models and leads to more efficiency in proofs about these kind of models, but also makes it possible to prove new results about the special cases which were not known before and to construct new models. Generalising Realizability and Heyting Models

### Under consideration for publication in Math. Struct. in Comp. Science Krivine’s Classical Realizability from a Categorical Perspective

, 2012

"... 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 choi ..."

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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 of this paper is to show how Krivine’s classical realizability can be understood as an instance of the categorical approach to realizability 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 realizability as can be found in (Krivine 2008). 1.

### Under consideration for publication in Math. Struct. in Comp. Science Krivine’s Classical Realizability from a Categorical Perspective

, 2012

"... 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 choi ..."

Abstract
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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 of this paper is to show how Krivine’s classical realizability can be understood as an instance of the categorical approach to realizability 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 realizability as can be found in (Krivine 2008). 1.

### Under consideration for publication in Math. Struct. in Comp. Science Krivine’s Classical Realizability from a Categorical Perspective

, 2010

"... 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 choi ..."

Abstract
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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 of this paper is to show how Krivine’s classical realizability can be understood as an instance of the categorical approach to realizability 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 realizability as can be found in (Krivine 2008). 1.