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Steve Awodey [52]Steven Awodey [3]Steven M. Awodey [1]
  1. Steve Awodey, Continuity and Logical Completeness.
    The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the..
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  2. Steve Awodey, Natural Models of Homotopy Type Theory.
    The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π, and intensional identity types Id, as used in (...)
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  3. Steve Awodey & Jesse Hughes, The Coalgebraic Dual of Birkhoff's Variety.
    ulations and show that they are definable by a trivial kind of coequation— namely, over one "color". We end with an example of a covariety which is not closed under bisimulations.
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  4. Peter T. Johnstone & Steve Awodey, Methodology.
    Notices Amer. Math. Sac. 51, 2004). Logically, such a "Grothendieck topos" is something like a universe of continuously variable sets. Before long, however, F.W. Lawvere and M. Tierney provided an elementary axiomatization..
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  5. Steve Awodey, Type Theory and Homotopy.
    of type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is also employed extensively as a framework for the development of high-level programming languages, in virtue of its combination of expressive strength and desirable proof-theoretic properties [NPS90, Str91]. In addition to simple types A, B, . . . and their terms x : A b(x) : B, the theory also has dependent types x : (...)
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  6. Steve Awodey & Michael A. Warren, Homotopy Theoretic Models of Identity Types.
    Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired (...)
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  7. Jesse Hughes, Steve Awodey, Dana Scott, Jeremy Avigad & Lawrence Moss, A Study of Categorres of Algebras and Coalgebras.
    This thesis is intended t0 help develop the theory 0f coalgebras by, Hrst, taking classic theorems in the theory 0f universal algebras amd dualizing them and, second, developing an interna] 10gic for categories 0f coalgebras. We begin with an introduction t0 the categorical approach t0 algebras and the dual 110tion 0f coalgebras. Following this, we discuss (c0)a,lg€bra.s for 2. (c0)monad and develop 2. theory 0f regular subcoalgebras which will be used in the interna] logic. We also prove that categories 0f (...)
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  8. Steve Awodey (forthcoming). Carnap and the Invariance of Logical Truth. Synthese:1-12.
    The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...)
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  9. Steve Awodey (2013). First-Order Logical Duality. Annals of Pure and Applied Logic 164 (3):319-348.
    From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this (...)
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  10. Steve Awodey (2013). Structuralism, Invariance, and Univalence. Philosophia Mathematica 22 (1):nkt030.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  11. Steve Awodey (2012). Explicating "Analytic". In Pierre Wagner (ed.), Carnap's Ideal of Explication and Naturalism. Palgrave Macmillan.
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  12. Georges Gonthier, Martin Ziegler, Steve Awodey, George Barmpalias & Lev D. Beklemishev (2012). Barcelona, Catalonia, Spain July 11–16, 2011. Bulletin of Symbolic Logic 18 (3).
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  13. Steve Awodey (2010). Category Theory. Oup Oxford.
    A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems , as well as numerous examples and exercises.
     
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  14. Steve Awodey & A. W. Carus (2010). Gödel and Carnap. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  15. Steve Awodey (2009). From Sets to Types to Categories to Sets. Philosophical Explorations.
    Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...)
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  16. Steve Awodey & A. W. Carus (2009). From Wittgenstein's Prison to the Boundless Ocean : Carnap's Dream of Logical Syntax. In Pierre Wagner (ed.), Carnap's Logical Syntax of Language. Palgrave Macmillan.
  17. Steve Awodey (2008). A Brief Introduction to Algebraic Set Theory. Bulletin of Symbolic Logic 14 (3):281-298.
    This brief 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, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of (...)
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  18. Steve Awodey (2008). Topology and Modality: The Topological Interpretation of First-Order Modal Logic: Topology and Modality. Review of Symbolic Logic 1 (2):146-166.
    As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.
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  19. Steven Awodey & Andrej Bauer (2008). Sheaf Toposes for Realizability. Archive for Mathematical Logic 47 (5):465-478.
    Steve Awodey and Audrej Bauer. Sheaf Toposes for Realizability.
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  20. Steve Awodey (2007). In Memoriam: Saunders Mac Lane, 1909-2005. Bulletin of Symbolic Logic 13 (1):115-119.
  21. Steve Awodey, Carsten Butz & Alex Simpson (2007). Relating First-Order Set Theories and Elementary Toposes. 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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  22. Steve Awodey, Carston Butz, Alex Simpson & Thomas Streicher, Relating First-Order Set Theories, Toposes and Categories of Classes.
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  23. Steve Awodey & Thomas Streicher (2007). Relating First-Order Set Theories and Elementary Toposes. 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 . 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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  24. Yiannis Moschovakis, Richmond H. Thomason, Steffen Lempp, Steve Awodey, Jean-Pierre Marquis & William Tait (2007). The Palmer House Hilton Hotel, Chicago, Illinois April 19–21, 2007. Bulletin of Symbolic Logic 13 (4).
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  25. Steve Awodey (2006). Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 139--149.
    The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets . In this (...)
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  26. Steve Awodey, Raf Cluckers, Ilijas Farah, Solomon Feferman, Deirdre Haskell, Andrey Morozov, Vladimir Pestov, Andre Scedrov, Andreas Weiermann & Jindrich Zapletal (2006). Stanford University, Stanford, CA March 19–22, 2005. Bulletin of Symbolic Logic 12 (1).
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  27. Steve Awodey, Henrik Forssell & Michael A. Warren, Algebraic Models of Sets and Classes in Categories of Ideals.
    We introduce a new sheaf-theoretic 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.
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  28. Steve Awodey (2005). Johnstone Peter T.. Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides, Vols. 43, 44. Oxford University Press, Oxford, 2002, Xxii+ 1160 Pp. [REVIEW] Bulletin of Symbolic Logic 11 (1):65-69.
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  29. Steve Awodey, Notes on Algebraic Set Theory.
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  30. Steve Awodey & A. W. Carus, The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism From Tractatus on Logical Syllogism.
    Steve Awodey and A. W. Carus. The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus on Logical Syllogism.
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  31. P. T. Johnstone & Steve Awodey (2005). REVIEWS-Sketches of an Elephant: A Topos Theory Compendium. Bulletin of Symbolic Logic 11 (1):65-69.
     
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  32. Erich H. Reck & Steve Awodey (2005). Frege's Lectures on Logic: Carnap's Student Notes, 1910-1914. Bulletin of Symbolic Logic 11 (3):445-447.
     
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  33. Steve Awodey (2004). An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’. Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  34. Steve Awodey, Book Review: Sketches of an Elephant. [REVIEW]
    Steve Awodey. Book Review: Sketches of an Elephant.
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  35. Steve Awodey & Jonas Eliasson (2004). Ultrasheaves and Double Negation. Notre Dame Journal of Formal Logic 45 (4):235-245.
    Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters—the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.
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  36. Steve Awodey & Henrik Forssell, Algebraic Models of Intuitionistic Theories of Sets and Classes.
    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 (...)
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  37. Steve Awodey & Michael A. Warren, Predicative Algebraic Set Theory.
    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 (...)
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  38. Jonas Eliasson & Steve Awodey (2004). Ultrasheaves and Double Negation. Notre Dame Journal of Formal Logic 45 (4):235-245.
    Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters - the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.
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  39. Steve Awodey & Jess Hughes, Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
    Steve Awodey and Jesse Hughes. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
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  40. Steve Awodey & Erich H. Reck, Completeness and Categoricity, Part I: 19th Century Axiomatics to 20th Century Metalogic.
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
     
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  41. Steve Awodey & Erich H. Reck, Completeness and Categoricty, Part II: 20th Century Metalogic to 21st Century Semantics.
    This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
     
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  42. Steve Awodey & Erich H. Reck (2002). Completeness and Categoricity. Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic. History and Philosophy of Logic 23 (1):1-30.
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  43. Steve Awodey & Erich H. Reck (2002). Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-First-Century Semantics. History and Philosophy of Logic 23 (2):77-94.
    This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  44. Itay Neeman, Alexander Leitsch, Toshiyasu Arai, Steve Awodey, James Cummings, Rod Downey & Harvey Friedman (2002). 2001 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'01. Bulletin of Symbolic Logic 8 (1):111-180.
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  45. Steve Awodey & Andrej Bauer, Propositions as [Types].
    Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using (...)
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  46. Steve Awodey & A. W. Carus, How Carnap Could Have Replied to Gödel.
    Steve Awodey and A. W. Carus. How Carnap Could Have Replied to Gödel.
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  47. Steve Awodey & Erich H. Reck, Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.
    Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.
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  48. Steve Awodey, Lars Birkedal & Dana Scott, Local Realizability Toposes and a Modal Logic for Computability.
    This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...)
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  49. Steve Awodey & Jesse Hughes, The Coalegebraic Dual of Birkoff's Variety Theorem.
    Steve Awodey and Jesse Hughes. The Coalegebraic Dual of Birkoff's Variety Theorem.
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  50. Steve Awodey, Topological Representation of the Lambda-Calculus.
    The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed categories, as well (...)
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