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Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
Aspects of predicative algebraic set theory I: Exact Completion
 Ann. Pure Appl. Logic
"... This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on ..."
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This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on
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
A general construction of internal sheaves in algebraic set theory. Preliminary version available at [3
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.
LAWVERETIERNEY SHEAVES IN ALGEBRAIC SET THEORY
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.
COHEN AND SET THEORY
"... Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ..."
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Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. That is, he established that Con(ZF) implies Con(ZF+¬AC) and Con(ZFC) implies Con(ZFC+¬CH). Already prominent as an analyst, Cohen had ventured into set theory with fresh eyes and an openmindedness about possibilities. These results delimited ZF and ZFC in terms of the two fundamental issues at the beginnings of set theory. But beyond that, Cohen’s proofs were the inaugural examples of a new technique, forcing, which was to become a remarkably general and flexible method for extending models of set theory. Forcing has strong intuitive underpinnings and reinforces the notion of set as given by the firstorder ZF axioms with conspicuous uses of Replacement and Foundation. If Gödel’s construction of L had launched set theory as a distinctive field of mathematics, then Cohen’s forcing began its transformation into a modern, sophisticated one. The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. Soon, ZFC became quite unlike Euclidean geometry and much like group theory, with a wide range of models of set theory being investigated for their own sake. Set theory had undergone a seachange, and with the subject so enriched, it is difficult to convey the strangeness of it. Received April 24, 2008. This is the full text of an invited address given at the annual meeting of the Association
Abstract Nonwellfounded trees in categories
"... nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, usi ..."
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nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with Mtypes under various topostheoretic constructions; namely, slicing, formation of coalgebras (for a cartesian comonad), and sheaves for an internal site. 1
Bireflectivity
, 1999
"... Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempoten ..."
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Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempotent natural transformations on idA. In the special case that A is a presheaf category, we characterise them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [C op; Set] and deduce results relating its finite product structure with the monoidal structure of [C op; Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relations in a topos A, and a characterisation of bireflection functors in terms of modules they define.