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The author introduces Lawvere and Tierney's concept of topos theory, a striking development in category theory that unites a number of important but seemingly diverse notions from algebraic geometry, set theory, and intuitionistic logic. Topos theory has led to the forging of surprising new links between classical and constructive mathematics. Bell presents toposes as the models of theories--the so-called local set theories--formulated within a typed intuitionistic logic.

Category theory. Homological algebra --- Toposes --- Set theory --- Logic, symbolic and mathematical --- Logic, Symbolic and mathematical --- Topoi (Mathematics) --- Categories (Mathematics) --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Mathematics --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Syllogism --- Logic, Symbolic and mathematical. --- Set theory. --- Toposes. --- Topos (mathématiques) --- Logique mathématique --- Théorie des ensembles --- Logique mathématique --- Théorie des ensembles --- Topos (mathématiques) --- Catégories (mathématiques)

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Independence (Mathematics) --- 510.2 --- Logic, Symbolic and mathematical --- Foundations of mathematics --- Algebra, Boolean. --- Axiomatic set theory. --- Model theory. --- Independence (Mathematics). --- 510.2 Foundations of mathematics --- Algebra, Boolean --- Axiomatic set theory --- Model theory --- Axioms --- Set theory --- Boolean algebra --- Boole's algebra --- Algebraic logic

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This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory (also known as higher-order intuitionistic logic), an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined.

Logic, Symbolic and mathematical. --- Type theory. --- Set theory. --- Logic, Symbolic and mathematical --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets

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Mathematical logic

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Mathematical logic