Goldblatt Topoi

The Categorial Analysis of Logic
2. Auflage 2014
ISBN: 978-1-4832-9921-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 98, 565 Seiten, Web PDF

Reihe: Studies in Logic and the Foundations of Mathematics

The first of its kind, this book presents a widely accessible exposition of topos theory, aimed at the philosopher-logician as well as the mathematician. It is suitable for individual study or use in class at the graduate level (it includes 500 exercises). It begins with a fully motivated introduction to category theory itself, moving always from the particular example to the abstract concept. It then introduces the notion of elementary topos, with a wide range of examples and goes on to develop its theory in depth, and to elicit in detail its relationship to Kripke's intuitionistic semantics, models of classical set theory and the conceptual framework of sheaf theory (``localization'' of truth). Of particular interest is a Dedekind-cuts style construction of number systems in topoi, leading to a model of the intuitionistic continuum in which a ``Dedekind-real'' becomes represented as a ``continuously-variable classical real number''.The second edition contains a new chapter, entitled Logical Geometry, which introduces the reader to the theory of geometric morphisms of Grothendieck topoi, and its model-theoretic rendering by Makkai and Reyes. The aim of this chapter is to explain why Deligne's theorem about the existence of points of coherent topoi is equivalent to the classical Completeness theorem for ``geometric'' first-order formulae.

Autoren/Hrsg.

Goldblatt, R.

Inhaltsverzeichnis

1;Front Cover;1
2;Topoi: The Categorial Analysis of Logic;4
3;Copyright Page;5
4;Table of Contents;16
5;Dedication;6
6;PREFACE;10
7;PREFACE TO SECOND EDITION;15
8;PROSPECTUS;18
9;CHAPTER 1. MATHEMATICS = SET THEORY?;23
9.1;1. Set theory;23
9.2;2. Foundations of mathematics;30
9.3;3. Mathematics as set theory;31
10;CHAPTER 2. WHAT CATEGORIES ARE;34
10.1;1. Functions are sets?;34
10.2;2. Composition of functions;37
10.3;3. Categories: first examples;40
10.4;4. The pathology of abstraction;42
10.5;5. Basic examples;43
11;CHAPTER 3. ARROWS INSTEAD OF EPSILON;54
11.1;1. Monic arrows;54
11.2;2. Epic arrows;56
11.3;3. Iso arrows;56
11.4;4. Isomorphic objects;58
11.5;5. Initial objects;60
11.6;6. Terminal objects;61
11.7;7. Duality;62
11.8;8. Products;63
11.9;9. Co-products;71
11.10;10. Equalisers;73
11.11;11. Limits and co-limits;75
11.12;12. Co-equalisers;77
11.13;13. The
pullback;80
11.14;14. Pushouts;85
11.15;15. Completeness;86
11.16;16. Exponentiation;87
12;CHAPTER 4. INTRODUCING TOPOI;92
12.1;1. Subobjects;92
12.2;2. Classifying subobjects;96
12.3;3. Definition of topos;101
12.4;4. First examples;102
12.5;5. Bundles and sheaves;105
12.6;6. Monoid actions;117
12.7;7. Power objects;120
12.8;8. O
and comprehension;124
13;CHAPTER 5. TOPOS STRUCTURE: FIRST STEPS;126
13.1;1. Monies equalise;126
13.2;2. Images of arrows;127
13.3;3. Fundamental facts;131
13.4;4. Extensionality and bivalence;132
13.5;5. Monies and epics by elements;140
14;CHAPTER 6. LOGIC CLASSICALLY CONCEIVED;142
14.1;1. Motivating topos logic;142
14.2;2. Propositions and truth-values;143
14.3;3. The prepositional calculus;146
14.4;4. Boolean algebra;150
14.5;5. Algebraic semantics;152
14.6;6. Truth-functions as arrows;153
14.7;7.
E-semantics;157
15;CHAPTER 7. ALGEBRA OF SUBOBJECTS;163
15.1;1. Complement, intersection, union;163
15.2;2. Sub(d) as a lattice;168
15.3;3. Boolean topoi;173
15.4;4. Internal vs. external;176
15.5;5. Implication and its implications;179
15.6;6. Filling two gaps;183
15.7;7. Extensionality revisited;185
16;CHAPTER 8. INTUITIONISM AND ITS LOGIC;190
16.1;1. Constructivist philosophy;190
16.2;2. Heyting's calculus;194
16.3;3. Heyting algebras;195
16.4;4. Kripke semantics;204
17;CHAPTER 9. FUNCTORS;211
17.1;1. The concept of functor;211
17.2;2. Natural transformations;215
17.3;3. Functor categories;219
18;CHAPTER 10. SET CONCEPTS AND VALIDITY;228
18.1;1. Set concepts;228
18.2;2. Heyting algebras in P;230
18.3;3. The subobject classifier in
Setp;232
18.4;4. The truth arrows;238
18.5;5. Validity;240
18.6;6. Applications;244
19;CHAPTER 11. ELEMENTARY TRUTH;247
19.1;1. The idea of a first-order
language;247
19.2;2. Formal language and
semantics;251
19.3;3. Axiomatics;254
19.4;4. Models in a topos;255
19.5;5. Substitution and soundness;266
19.6;6. Kripke models;273
19.7;7. Completeness;281
19.8;8. Existence and free logic;283
19.9;9. Heyting-valued sets;291
19.10;10. High-order logic;303
20;CHAPTER 12. CATEGORIAL SET THEORY;306
20.1;1. Axioms of choice;307
20.2;2. Natural numbers objects;318
20.3;3. Formal set theory;322
20.4;4. Transitive sets;330
20.5;5. Set-objects;337
20.6;6. Equivalence of models;345
21;CHAPTER 13. ARITHMETIC;349
21.1;1. Topoi as foundations;349
21.2;2. Primitive recursion;352
21.3;3. Peano postulates;364
22;CHAPTER 14. LOCAL TRUTH;376
22.1;1. Stacks and sheaves;376
22.2;2. Classifying stacks and sheaves;385
22.3;3. Grothendieck
topoi;391
22.4;4. Elementary sites;395
22.5;5. Geometric modality;398
22.6;6. Kripke–Joyal
semantics;403
22.7;7. Sheaves as complete
O-sets;405
22.8;8. Number systems as sheaves;430
23;CHAPTER 15. ADJOINTNESS AND QUANTIFIERS;455
23.1;1. Adjunctions;455
23.2;2. Some adjoint situations;459
23.3;3. The fundamental theorem;466
23.4;4. Quantifiers;470
24;CHAPTER 16. LOGICAL GEOMETRY;475
24.1;1. Preservation and reflection;476
24.2;2. Geometric morphisms;480
24.3;3. Internal logic;500
24.4;4. Geometric logic;510
24.5;5. Theories as sites;521
25;REFERENCES;538
26;CATALOGUE OF NOTATION;548
27;INDEX OF DEFINITIONS;558