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22003 european summer meeting of the association for symbolic logic logic colloquim'03Bulletin of Symbolic Logic 10 (2). 2004.
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23Review: Andrej Scedrov, Forcing and Classifying Topoi (review)Journal of Symbolic Logic 50 (3): 852-853. 1985.
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66Notices 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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Finitary SketchesJournal of Symbolic Logic 62 (3): 699-707. 1997.Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by $\sigma$-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equ…Read more
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2Sketches of an Elephant: 2 Volume SetOxford University Press UK. 2002.Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the ov…Read more
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4Sketches of an Elephant: Volume 2Oxford University Press UK. 2002.Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.
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7Sketches of an Elephant: A Topos Theory Compendium: 2 Volume SetOxford University Press UK. 2002.Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the ov…Read more
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8Sketches of an Elephant: A Topos Theory Compendium: Volume 2Oxford University Press UK. 2002.Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.
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14Complemented sublocales and open mapsAnnals of Pure and Applied Logic 137 (1-3): 240-255. 2006.We show that a morphism of locales is open if and only if all its pullbacks are skeletal in the sense of [P.T. Johnstone, Factorization theorems for geometric morphisms, II, in: Categorical Aspects of Topology and Analysis, in: Lecture Notes in Math., vol. 915, Springer-Verlag, 1982, pp. 216–233], i.e. pulling back along them preserves denseness of sublocales . This result may be viewed as the ‘dual’ of the well-known characterization of proper maps as those which are stably closed. We also inve…Read more
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39Steven Vickers. Topology via logic. Cambridge tracts in theoretical computer science, no. 5. Cambridge University Press, Cambridge etc. 1989, xiii + 200 pp (review)Journal of Symbolic Logic 56 (3): 1101-1102. 1991.
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106Review: Steven Vickers, Topology via Logic (review)Journal of Symbolic Logic 56 (3): 1101-1102. 1991.
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16Ščedrov Andrej. Forcing and classifying topoi. Memoirs of the American Mathematical Society, no. 295. American Mathematical Society, Providence 1984, x + 93 pp (review)Journal of Symbolic Logic 50 (3): 852-853. 1985.
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49Finitary sketchesJournal of Symbolic Logic 62 (3): 699-707. 1997.Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by σ-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalenc…Read more
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98Notes on logic and set theoryCambridge University Press. 1987.A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal a…Read more
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37Sketches of an Elephant: A Topos Theory Compendium, Volume 1Clarendon Press. 2002.Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.
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25What do Freyd’s Toposes Classify?Logica Universalis 7 (3): 335-340. 2013.We describe a method for presenting (a topos closely related to) either of Freyd’s topos-theoretic models for the independence of the axiom of choice as the classifying topos for a geometric theory. As an application, we show that no such topos can admit a geometric morphism from a two-valued topos satisfying countable dependent choice
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40Classifying toposes for first-order theoriesAnnals of Pure and Applied Logic 91 (1): 33-58. 1998.By a classifying topos for a first-order theory , we mean a topos such that, for any topos models of in correspond exactly to open geometric morphisms → . We show that not every first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a…Read more
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12Jaap van Oosten. Realizability: an introduction to its categorical side. Studies in Logic and the Foundations of Mathematics, vol. 152. Elsevier Science, Amsterdam, 2008, 328 pp (review)Bulletin of Symbolic Logic 16 (3): 407-409. 2010.
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2Noble Cause Police Corruption: Suggestions for TrainingInternational Journal of Applied Philosophy 16 (2): 249-264. 2002.This essay confronts police corruption historically and conceptually, isolating noble cause corruption as a neglected yet powerful motivator of corrupt police behavior. Noble cause corruption is defined in some detail and several specific suggestions are made regarding police training programs to address the issue.