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17
Presheaf models of constructive set theories
, 2004
"... Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic S ..."
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Cited by 16 (4 self)
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Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects
Algebraic models of intuitionistic theories of sets and classes. Theory and applications of categories
"... Abstract. This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a g ..."
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Cited by 13 (7 self)
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Abstract. This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in [2] by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three. 1.
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 8 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
Predicative algebraic set theory
 Theory and Applications of Categories
"... Abstract. In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certa ..."
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Cited by 7 (1 self)
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Abstract. In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of ideals. The purpose of this paper is to generalize the machinery and results developed by Awodey, Butz, Simpson and Streicher in [Awodey et al, 2004] to the predicative case. Specifically, in ibid. it was shown that: 1. every category of classes contains a model of the intuitionistic, elementary set theory BIST,
Relating firstorder set theories and elementary toposes
 BULLETIN OF SYMBOLIC LOGIC
, 2007
"... We show how to interpret the language of firstorder 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. ..."
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Cited by 6 (4 self)
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We show how to interpret the language of firstorder 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 firstorder set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic ZermeloFraenkel set theory (IZF).
A unified approach to algebraic set theory
 the proceedings of the Logic Colloquium 2006, arXiv:0710.3066
, 2007
"... ..."
Relating firstorder set theories, toposes and categories of classes
 In preparation
, 2006
"... This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingst ..."
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Cited by 4 (4 self)
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This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingstyle interpretation of the language of firstorder set theory in the topos is given, which conservatively extends the internal logic of the topos. Since every topos is equivalent to one carrying a dssi, the language of firstorder has a forcing interpretation in every elementary topos. We prove that the set theory BIST+ Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that every cocomplete topos and every realizability topos can be endowed (up to equivalence) with such a superdirected structural system of inclusions. This provides a uniform explanation for why such “realworld ” toposes model Separation. A large part of the paper is devoted to an alternative notion of categorytheoretic model for BIST, which, following the general approach of Joyal and Moerdijk’s Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with ∗Corresponding author. 1Previously, lecturer at HeriotWatt University (2000–2001), and the IT University of
Categorical models of intuitionistic theories of sets and classes
, 2004
"... The thesis consists of three sections, developing models of intuitionistic set theory in suitable categories. First, the categorical framework in which models are constructed is reviewed, and the theory of all such models, called Basic Intuitionistic Set Theory (BIST), is stated; second, we give a n ..."
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Cited by 3 (2 self)
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The thesis consists of three sections, developing models of intuitionistic set theory in suitable categories. First, the categorical framework in which models are constructed is reviewed, and the theory of all such models, called Basic Intuitionistic Set Theory (BIST), is stated; second, we give a notion of an ideal over a category, with which one can build a model of BIST in which a given topos occurs as the sets; and third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST.