Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Fraenkel set theory. – This book studies Cantor's original development of set theory and its main subsequent development in the following thirty years. In his discussion of Cantor's work, Hallett addresses himself to three main questions. How did “set” become the fundamental notion in Cantor's theory ? What was Cantor's own conception of set ? What effect did Cantor's philosophical ideas have on the shape of his own theory and on what came later ? Part 2 of the book considers the extent to which modern set theory is properly to be seen as the axiomatic development (notably by Zermelo, Fraenkel and von Neumann) of Cantor's original conception. The universality of set construction can lead to paradoxes. Limitations of size as a basis for consistent elucidation of the set concept is an underlying theme of this work. Hallett's book makes an important contribution, both for the author's own insights and for his careful exposition of historical development, with detailed references and extensive quotation from the literature, including work by Dedekind, Frege, Russell, Jourdain, Miramanoff, Hessenberg, W.H. and G.C. Young, Hausdorff, Kuratowski, Bernays, Gödel, and more. – Part 1 : «The Cantorian origins of set theory»; – Part 2 : «The limitation of size argument and axiomatic set theory».