Topoi: The Categorial Analysis of Logic

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