Sets for Mathematics
Edition: 1st edition
Publisher: Cambridge University Press
Published: December 2002
Language: eng
Format: Paperback
Number of pages: 250
ISBN-13: 9780521010603
Subjects: Set Theory; Mathematics; Science/Mathematics; Mathematics / Combinatorics
First sentence: Let us discuss the idea of abstract constant sets and the mappings between them in order to have a picture of this, our central example, before formalizing a mathematical definition.
Description: Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. This book, first published in 2003, uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms which express universal properties of sums, products, mapping sets, and natural number recursion. The distinctive features of Cantorian abstract sets, as contrasted with the variable and cohesive sets of geometry and analysis, are made explicit and taken as special axioms. Functor categories are introduced in order to model the variable sets used in geometry, and to illustrate the failure of the axiom of choice. An appendix provides an explicit introduction to necessary concepts from logic, and an extensive glossary provides a window to the mathematical landscape.