Bulletin of Symbolic Logic

Research Article

Zermelo and Set Theory

Akihiro Kanamori *

Department of Mathematics, Boston University, Boston, MA 02215, E-mail: , aki@math.bu.edu

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

(Received March 08 2004)

(Accepted June 02 2004)


*   This is a much expanded version of an invited address, on the occasion of the 50th anniversary of the death of Zermelo, at the 12th International Congress of Logic, Methodology and Philosophy of Science held August 7–13, 2003 at Oviedo, Spain, and the author gratefully thanks the organizers for the invitation. The author also gratefully acknowledges the generous support of the Dibner Institute for the History of Science and Technology, the hospitality of the Department of Pure Mathematics and Mathematical Statistics at Cambridge University, and helpful and valuable comments by Heinz-Dieter Ebbinghaus, Juliet Floyd, Volker Peckhaus, Gregory Taylor, and the referee.