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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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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,
Aspects of predicative algebraic set theory II: Realizability. Accepted for publication in Theoretical Computer Science
 In Logic Colloquim 2006, Lecture Notes in Logic
, 2009
"... This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how ..."
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This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how these predicative categories
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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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
Algebraic models of sets and classes in categories of ideals
 In preparation
, 2006
"... We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produc ..."
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We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories. 1
On the Constructive Dedekind Reals
 Log. Anal
, 2008
"... In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alon ..."
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In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class. 1
Part of the Philosophy Commons Recommended Citation
, 2006
"... . Algebraic models of sets and classes in categories of ideals ..."
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An Outline of Algebraic Set Theory
"... This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical ..."
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This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals ” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are
iii ACKNOWLEDGEMENTS
, 2008
"... Paul Sternberg, James R. Heath, and (above all) my advisor, Stephen Quake, for their advice and support. My scientific collaborators, David Bryder and Derrick Rossi, and their director, Irving Weissman, allowed me to apply novel singlecell analysis techniques to real biological problems. Stuart Kim ..."
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Paul Sternberg, James R. Heath, and (above all) my advisor, Stephen Quake, for their advice and support. My scientific collaborators, David Bryder and Derrick Rossi, and their director, Irving Weissman, allowed me to apply novel singlecell analysis techniques to real biological problems. Stuart Kim and Jan Vijg were guiding influences for the aging study, and Geoffrey Schiebinger helped out tremendously with the experimental work. I extend my appreciation to the members of the Quake lab—most especially, to Joshua Weinstein, who worked out the analytic solution to the digitalPCR responsecurve problem that appears in the appendix, and to Joshua Marcus, who gave freely of his time to get me started on microfluidics. I am also grateful to Eric Davidson for sharing his hardwon insights into the character of generegulatory networks, and Stuart Kauffman for stimulating conversations on the same subject. Both have written excellent books, which will shape our understanding for years to come. iv The problem of development has long been one of the key issues in biology. With stemcell
Sets, Categories and
"... First we introduce some basic theoretical issues that set the stage for subsequent accounts. Secondly, we touch upon some important issues: Settheory vs. category theory, various conceptions of sets, ..."
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First we introduce some basic theoretical issues that set the stage for subsequent accounts. Secondly, we touch upon some important issues: Settheory vs. category theory, various conceptions of sets,