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Alex Simpson [9]Alex K. Simpson [1]Alexander Russell Simpson [1]Alexander Thomas Simpson [1]
  1.  73
    Relating First-Order Set Theories, Toposes and Categories of Classes.Steve Awodey, Carsten Butz, Alex Simpson & Thomas Streicher - 2014 - Annals of Pure and Applied Logic 165 (2):428-502.
  2.  51
    Relating First-Order Set Theories and Elementary Toposes.Steve Awodey, Carsten Butz & Alex Simpson - 2007 - Bulletin of Symbolic Logic 13 (3):340-358.
    We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...)
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  3.  15
    Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory.Alex Simpson - 2004 - Annals of Pure and Applied Logic 130 (1-3):207-275.
    This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, (...)
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  4.  25
    Two Constructive Embedding‐Extension Theorems with Applications to Continuity Principles and to Banach‐Mazur Computability.Andrej Bauer & Alex Simpson - 2004 - Mathematical Logic Quarterly 50 (4‐5):351-369.
    We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful (...)
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  5.  6
    Axioms and (Counter)Examples in Synthetic Domain Theory.Jaap van Oosten & Alex K. Simpson - 2000 - Annals of Pure and Applied Logic 104 (1-3):233-278.
    An axiomatic treatment of synthetic domain theory is presented, in the framework of the internal logic of an arbitrary topos. We present new proofs of known facts, new equivalences between our axioms and known principles, and proofs of new facts, such as the theorem that the regular complete objects are closed under lifting . In Sections 2–4 we investigate models, and obtain independence results. In Section 2 we look at a model in de Modified realizability Topos, where the Scott Principle (...)
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  6.  45
    Measure, Randomness and Sublocales.Alex Simpson - 2012 - Annals of Pure and Applied Logic 163 (11):1642-1659.
    This paper investigates aspects of measure and randomness in the context of locale theory . We prove that every measure μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X . Furthermore, when μ is a finite measure with μ=M, the σ-locale X has a smallest σ-sublocale of measure M . In particular, when μ is a probability measure, X has a smallest σ-sublocale of measure 1. All (...)
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  7.  4
    Two Constructive Embedding-Extension Theorems with Applications to Continuity Principles and to Banach-Mazur Computability.Andrej Bauer & Alex Simpson - 2004 - Mathematical Logic Quarterly 50 (45):351-369.
    We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful (...)
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  8.  37
    Constructive Toposes with Countable Sums as Models of Constructive Set Theory.Alex Simpson & Thomas Streicher - 2012 - Annals of Pure and Applied Logic 163 (10):1419-1436.
  9.  25
    Copies of Books to Asl, Box 742, Vassar College, 124 Raymond Avenue, Poughkeepsie, Ny 12604, Usa. In a Review, a Reference “Jsl Xliii 148,” for Example, Refers Either to the Publication Reviewed on Page 148 of Volume 43 of the Journal, or to the Review Itself (Which Contains Full Bibliographical Information for the Reviewed Publication). Analogously, a Reference. [REVIEW]Anuj Dawar Colyvan, Steffen Lempp, Rahim Moosa, Ernest Schimmerling & Alex Simpson - 2013 - Bulletin of Symbolic Logic 19 (2).
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  10.  12
    Relating Topos Theory and Set Theory Via Categories of Classes.Steve Awodey, Alex Simpson & Thomas Streicher - unknown
    We investigate a certain system of intuitionistic set theory from three points of view: an elementary set theory with bounded separation, a topos with distinguished inclusions, and a category of classes with a system of small maps. The three presentations are shown to be equivalent in a strong sense.
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