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  1. Sori Lee & Jaap Van Oosten (2013). Basic Subtoposes of the Effective Topos. Annals of Pure and Applied Logic 164 (9):866-883.
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  2. Juliette Kennedy & Jaap van Oosten (2012). Preface. Annals of Pure and Applied Logic 163 (10):1359.
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  3. Jaap van Oosten (2011). Partial Combinatory Algebras of Functions. Notre Dame Journal of Formal Logic 52 (4):431-448.
    We employ the notions of "sequential function" and "interrogation" (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is that every realizability topos is a geometric quotient of a realizability topos on a total combinatory algebra.
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  4. Jaap van Oosten (2006). A General Form of Relative Recursion. Notre Dame Journal of Formal Logic 47 (3):311-318.
    The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the (...)
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  5. Jaap Van Oosten (2006). From Sets and Types to Topology and Analysis—Towards Practicable Foundations for Constructive Mathematics, Edited by Crosilla Laura and Schuster Peter, Oxford Logic Guides, Vol. 48. Clarendon Press, 2005, Xix+ 450 Pp. [REVIEW] Bulletin of Symbolic Logic 12 (4):611-612.
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  6. Claire Kouwenhoven-Gentil & Jaap van Oosten (2005). Algebraic Set Theory and the Effective Topos. Journal of Symbolic Logic 70 (3):879 - 890.
    Following the book Algebraic Set Theory from André Joyal and leke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).
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  7. Jaap van Oosten (2004). A Partial Analysis of Modified Realizability. Journal of Symbolic Logic 69 (2):421-429.
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  8. Jaap van Oosten (2004). Sheaves, Games, and Model Completions. Bulletin of Symbolic Logic 10 (2):216-218.
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  9. Lars Birkedal & Jaap van Oosten (2002). Relative and Modified Relative Realizability. Annals of Pure and Applied Logic 118 (1-2):115-132.
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  10. Jaap van Oosten & Harold Schellinx (2002). Preface. Annals of Pure and Applied Logic 114 (1-3):1-2.
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  11. Jaap van Oosten & Alex K. Simpson (2000). Axioms and (Counter)Examples in Synthetic Domain Theory. Annals of Pure and Applied Logic 104 (1-3):233-278.
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  12. Jaap van Oosten (1997). Extensional Realizability. Annals of Pure and Applied Logic 84 (3):317-349.
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  13. Jaap van Oosten (1996). Two Remarks on the Lifschitz Realizability Topos. Journal of Symbolic Logic 61 (1):70-79.
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  14. Jaap van Oosten (1994). Axiomatizing Higher-Order Kleene Realizability. Annals of Pure and Applied Logic 70 (1):87-111.
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  15. Jaap van Oosten (1991). A Semantical Proof of De Jongh's Theorem. Archive for Mathematical Logic 31 (2):105-114.
    In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 2 0 (...)
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  16. Jaap van Oosten (1991). Extension of Lifschitz' Realizability to Higher Order Arithmetic, and a Solution to a Problem of F. Richman. Journal of Symbolic Logic 56 (3):964-973.
    F. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic $\mathbf{HAH}$: $\forall X\lbrack\forall x(x \in X \vee \neg x \in X) \wedge \forall Y(\forall x(x \in Y \vee \neg x \in Y) \rightarrow \forall x(x \in X \rightarrow x \in Y) \vee \forall x \neg(x \in X \wedge x \in Y)) \rightarrow \exists n\forall x(x \in X \rightarrow x = n)\rbrack$, and if not, whether assuming Church's Thesis CT and (...)
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  17. Jaap Van Oosten (1990). Lifschitz' Realizability. Journal of Symbolic Logic 55 (2):805-821.
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