Topoi: The Categorial Analysis of Logic

Description

A classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally. Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry. Cover Title Preface Preface to the Second Edition Contents Prospectus 1. Mathematics=Set Theory? 1. Set Theory 2. Foundations of Mathematics 3. Mathematics as Set Theory 2. What Categories Are 1. Functions are Sets? 2. Compositions of Functions 3. Categories: First Examples 4. The Pathology of Abstraction 5. Basic Examples 3. Arrows Instead of Epsilon 1. Monic Arrows 2. Epic Arrows 3. Iso Arrows 4. Isomorphic Objects 5. Initial Objects 6. Terminal Objects 7. Duality 8. Products 9. Co-Products 10. Equalisers 11. Limits and Co-limits 12. Co-equalisers 13. The Pullback 14. Pushouts 15. Completeness 16. Exponentiation 4. Introducting Topoi 1. Subobjects 2. Classifying Subobjects 3. Definition of Topos 4. First Examples 5. Bundles and Sheaves 6. Monoid Actions 7. Power Objects 8. Ω and Comprehension 5. Topos Structure: First Steps 1. Monics Equalise 2. Images of Arrows 3. Fundamental Facts 4. Extensionality and Bivalence 5. Monics and Epics by Elements 6. Logic Classically Conceived 1. Motivating Topos Logic 2. Propositions and Truth-Values 3. The Propositional Calculus 4. Boolean Algebra 5. Algebraic Semantics 6. Truth Functions and Arrows Appendix 7. Algebra of Subobjects 1. Complement, Intersection, Union 2. Sub(d) as a Lattice 3. Boolean Topoi 4. Internal Vs. External 5. Implication and its Implications 6. Filling Two Gaps 7. Extensionality Revisited 8. Intuitionism and It's Logic 1. Constructivist Philosophy 2. Heyting's Calculus 3. Heyting Algebras 4. Kripke Semantics 9. Functors 1. The Concept of a Functor 2. Natural Transformations 3. Functor Categories 10. Set Concepts and Validity 1. Set Concepts 2. Heyting Algebras in P 3. The Subobject Classifier in Setᵖ 4. The Truth Arrows 5. Validity 6. Applications 11. Elementary Truth 1. The Idea of a First-Order Languange 2. Formal Languange and Semantics 3. Axiomatics 4. Models in a Topos 5. Substitution and Soundness 6. Kripke Models 7. Completeness 8. Existence and Free-Logic 9. Heyting Valued-Sets 10. Higher-Order Logic 12. Categorial Set Theory 1. Axioms of Choice 2. Natural Numbers Objects 3. Formal Set Theory 4. Transitive Sets 5. Set-Objects 6. Equivalence of Models 13. Arithmetic 1. Topoi as Foundations 2. Primitive Recursion 3. Peano Postulates 14. Local Truth 1. Stacks and Sheaves 2. Classifying Stacks and Sheaves 3. Grothendieck Topoi 4. Elementary Sites 5. Geometric Modality 6. Kripke-Joyal Semantics 7. Sheaves as Complete Ω-sets 8. Number Systems as Sheaves 15. Adjointness and Quantifiers 1. Adjunctions 2. Some Adjoint Situations 3. The Fundamental Theorem 4. Quantifiers 16. Logical Geometry 1. Preservation and Reflection 2. Geometric Morphisms 3. Internal Logic 4. Geometric Logic 5. Theories as Sites References Catalogue of Notation Index of Definitions