Topoi: The Categorial Analysis of Logic

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Elsevier, Jun 28, 2014 - Mathematics - 565 pages
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.
 

Contents

PROSPECTUS
1
CHAPTER 1 MATHEMATICS SET THEORY?
6
CHAPTER 2 WHAT CATEGORIES ARE
17
CHAPTER 3 ARROWS INSTEAD OF EPSILON
37
CHAPTER 4 INTRODUCING TOPOI
75
FIRST STEPS
109
CHAPTER 6 LOGIC CLASSICALLY CONCEIVED
125
CHAPTER 7 ALGEBRA OF SUBOBJECTS
146
CHAPTER 11 ELEMENTARY TRUTH
230
CHAPTER 12 CATEGORIAL SET THEORY
289
CHAPTER 13 ARITHMETIC
332
CHAPTER 14 LOCAL TRUTH
359
CHAPTER 15 ADJOINTNESS AND QUANTIFIERS
438
CHAPTER 16 LOGICAL GEOMETRY
458
REFERENCES
521
CATALOGUE OF NOTATION
531

CHAPTER 8 INTUITIONISM AND ITS LOGIC
173
CHAPTER 9 FUNCTORS
194
CHAPTER 10 SET CONCEPTS AND VALIDITY
211
INDEX OF DEFINITIONS
541
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