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Ieke Moerdijk [15]I. Moerdijk [8]
  1.  28
    Type theories, toposes and constructive set theory: predicative aspects of AST.Ieke Moerdijk & Erik Palmgren - 2002 - Annals of Pure and Applied Logic 114 (1-3):155-201.
    We introduce a predicative version of topos based on the notion of small maps in algebraic set theory, developed by Joyal and one of the authors. Examples of stratified pseudotoposes can be constructed in Martin-Löf type theory, which is a predicative theory. A stratified pseudotopos admits construction of the internal category of sheaves, which is again a stratified pseudotopos. We also show how to build models of Aczel-Myhill constructive set theory using this categorical structure.
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  2.  47
    Wellfounded trees in categories.Ieke Moerdijk & Erik Palmgren - 2000 - Annals of Pure and Applied Logic 104 (1-3):189-218.
    In this paper we present and study a categorical formulation of the W-types of Martin-Löf. These are essentially free term algebras where the operations may have finite or infinite arity. It is shown that W-types are preserved under the construction of sheaves and Artin gluing. In the proofs we avoid using impredicative or nonconstructive principles.
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  3.  20
    A model for intuitionistic non-standard arithmetic.Ieke Moerdijk - 1995 - Annals of Pure and Applied Logic 73 (1):37-51.
    This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice.
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  4.  55
    Minimal models of Heyting arithmetic.Ieke Moerdijk & Erik Palmgren - 1997 - Journal of Symbolic Logic 62 (4):1448-1460.
    In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain (...)
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  5.  18
    Aspects of predicative algebraic set theory I: Exact Completion.Benno van den Berg & Ieke Moerdijk - 2008 - Annals of Pure and Applied Logic 156 (1):123-159.
    This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...)
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  6.  8
    Minimal models of Heyting arithmetic.Ieke Moerdijk & Erik Palmgren - 1997 - Journal of Symbolic Logic 62 (4):1448-1460.
    In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain (...)
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  7.  13
    Derived rules for predicative set theory: an application of sheaves.Benno van den Berg & Ieke Moerdijk - 2012 - Annals of Pure and Applied Logic 163 (10):1367-1383.
  8.  20
    A completeness theorem for open maps.A. Joyal & I. Moerdijk - 1994 - Annals of Pure and Applied Logic 70 (1):51-86.
    This paper provides a partial solution to the completeness problem for Joyal's axiomatization of open and etale maps, under the additional assumption that a collection axiom holds.
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  9.  41
    Sheaf models for choice sequences.Gerrit Van Der Hoeven & Ieke Moerdijk - 1984 - Annals of Pure and Applied Logic 27 (1):63-107.
  10.  62
    Heine-borel does not imply the Fan theorem.Ieke Moerdijk - 1984 - Journal of Symbolic Logic 49 (2):514-519.
  11.  11
    Downey, R., Gasarch, W. and Moses, M., The structure.S. D. Friedman, W. G. Handley, S. S. Wainer, A. Joyal, I. Moerdijk, L. Newelski, F. van Engelen & J. van Oosten - 1994 - Annals of Pure and Applied Logic 70 (1):287.
  12.  45
    Sets, Topoi and Intuitionism.I. Moerdijk - 1998 - Philosophia Mathematica 6 (2):169-177.
    This paper aims to give an informal introduction to the ways in which a topos can be viewed as an intuitionistic universe of sets. In particular, it is explained how infinitesimal real numbers and various types of ordinal numbers arise in this context.
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  13.  43
    The axiom of multiple choice and models for constructive set theory.Benno van den Berg & Ieke Moerdijk - 2014 - Journal of Mathematical Logic 14 (1):1450005.
    We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory. In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as (...)
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  14. On choice sequences determined by spreads.Gerritder Hoeven & Ieke Moerdijk - 1984 - Journal of Symbolic Logic 49 (3):908 - 916.
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  15.  12
    Abrahamson, KA, Downey, RG and Fellows, MR.R. Banacb, H. Barendregt, J. A. Bergstra, J. V. Tucker, J. Brendle, I. Moerdijk, E. Palmgren, J. I. Seiferas, A. R. Meyer & J. Terlouw - 1995 - Annals of Pure and Applied Logic 73 (1):327.
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  16.  38
    An elementary definability theorem for first order logic.C. Butz & I. Moerdijk - 1999 - Journal of Symbolic Logic 64 (3):1028-1036.
  17.  9
    Nijmegen, The Netherlands July 27–August 2, 2006.Rodney Downey, Ieke Moerdijk, Boban Velickovic, Samson Abramsky, Marat Arslanov, Harvey Friedman, Martin Goldstern, Ehud Hrushovski, Jochen Koenigsmann & Andy Lewis - 2007 - Bulletin of Symbolic Logic 13 (2).
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  18.  16
    On the Freyd cover of a topos.Ieke Moerdijk - 1983 - Notre Dame Journal of Formal Logic 24 (4):517-526.
  19.  57
    Compositionality and the analysis of anaphora.Fred Landman & Ieke Moerdijk - 1983 - Linguistics and Philosophy 6 (1):89 - 114.
  20.  38
    On Choice Sequences Determined by Spreads.Gerrit van der Hoeven & Ieke Moerdijk - 1984 - Journal of Symbolic Logic 49 (3):908 - 916.
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  21.  12
    Jonahan Chapman and Frederick Rowbottom. Relative category theory and geometric morphisms. A logical approach. Oxford logic guides, no. 16., Clarendon press, Oxford University Press, Oxford and New York1992, xi + 263 pp. [REVIEW]I. Moerdijk - 1995 - Journal of Symbolic Logic 60 (2):694-695.
  22.  10
    Review: Jonathan Chapman, Frederick Rowbottom, Relative Category Theory and Geometric Morphisms. A Logical Approach. [REVIEW]I. Moerdijk - 1995 - Journal of Symbolic Logic 60 (2):694-695.