Mathematics = Set Theory? What Categories Are. Arrows Instead of Epsilon. Introducing Topoi. Topos Structure: First Steps. Logic Classically Conceived. Algebra of Subobjects. Intuitionism and its Logic. Functors. Set Concepts and Validity. Elementary Truth. Categorical Set Theory. Arithmetic. Local Truth. Adjointness and Quantifiers. Logical Geometry.
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 aDedekind-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.
- No. of pages:
- © North Holland 1984
- 1st February 1984
- North Holland
- eBook ISBN:
@qu:...an excellent expository text on elementary topoi... Dr. Goldblatt is completely successful in achieving his aim to introduce the reader to the notion of a topos and to explain what its implications are for logic and the foundations of mathematics. @source:Mathematical Chronicle @qu:The book is very clearly written... an excellent place to start finding out about topos theory. @source:Mathematical Review