Co-authored by one of the founders of category theory, Sets for Mathematics is implicitly a triad of books: it is, first, a textbook on what is traditionally called "naive set theory''; second, a well-written introduction to category theory with the category of sets as guiding principle; and third, a category-theoretic manifesto against traditional views of set theory, logic and the foundations of mathematics.
The authors present the elementary theory of sets and functions from the very basics to Cantor's theorem of the uncountability of the real line, in the style of category theory.
Between the lines and yet overtly, they profess their position on foundational matters: they embrace a categorial viewpoint (with object and morphism as basic notions) and renounce traditional foundations of mathematics (with set and element as basic notions and the cumulative hierarchy of sets as global structure). Thus, despite the title, this is not a book on sets or set theory but a book on category theory that happens to talk mainly about the category of sets.
Sets for Mathematics would be most useful in advanced undergraduate or graduate courses on category theory. With a firm grasp of naive set theory, advanced students could learn category theory while encountering a new view of naive set theory and a non-standard view of the foundations of mathematics.
Benedikt Löwe is Universitair Docent, Institute for Logic, Language and Computation, University of Amsterdam, Netherlands.
Sets for Mathematics. First edition
Author - F. William Lawvere and Robert Rosebrugh
Publisher - Cambridge University Press
Pages - 261
Price - £50.00 and £19.95
ISBN - 0 521 80444 2 and 01060 8