This is the first of two volumes containing Ernst Zermelo’s papers. It covers his work on set theory and the foundations of mathematics; the second volume will cover his work on physics. This one also has a fine short biography of Zermelo by Ebbinghaus, who has also recently published a full-length biography of him (Ebbinghaus 2007).
The papers are given in their original German with a facing translation into English. There are also extensive introductory notes, sometimes longer than the papers they introduce, which set the historical scene and bring out the importance of the papers, including, when appropriate, their imperfections.
In the present connection Zermelo is well-known for his argument that every set can be well-ordered. In 1904, during his time in Göttingen, Zermelo wrote up a proof that every set can be well-ordered, which he based on an explicitly formulated axiom of choice, and took it to Hilbert .
As Gregory Moore showed in (Moore 1982) Hilbert put Zermelo’s paper into Mathematische Annalen and the resulting furore about the implications for mathematics, especially set theory and the axiom of choice, changed the subject forever. Challenged to defend the axiom of choice Zermelo entirely reasonably pointed out that it had been used, explicitly and tacitly, quite often in mathematics, and it did not seem right to question it now. When the criticisms did not die down he took stock of his position and in 1908 published two articles. The first is a reply to his critics, where, as Moore showed, he successfully turned the arguments of his critics against each other. The second contains his axioms for set theory, which he intended to shore up his proof of the well-ordering principle. These axioms were also criticised, but eventually, in the improved form in which Fraenkel put them, they became the most widely accepted formulation for the foundations of the subject for many years.
Less well-known is what Zermelo did next. There is an interesting, if ultimately unsuccessful, attempt to apply set theory to the game of chess, a paper on measure theory, and some papers from 1929–1931 when Zermelo briefly returned to the foundations of mathematics and was preparing the edition of Cantor’s Collected Papers. In these papers he presented the idea of the cumulative hierarchy of sets, based on a collection of axioms that, after Skolem re-wrote them, became the usual first-order Zermelo-Fraenkel axioms with Choice (ZFC).
In these years Zermelo felt himself called upon to defend mathematics from Skolem and Gödel, and inevitably he lost. For much of his life he had been an isolated figure, handicapped by ill-health, and in the 1930s he cut a lonely figure hoping to establish the truth of mathematics when formalisms stood all around. Yet, as Ebbinghaus records, with the support of his much younger wife Zermelo lived long enough to survive the Second World War and be recognised by the next generation of German mathematicians for his major work.
Zermelo had apparently attempted to persuade Springer to publish his collected works in 1949, but the economic times were not suitable. The volume they have published in 2010 is one of their handsome editions, well bounded and attractively printed. The three editors, Hans-Dieter Ebbinghaus, Craig Fraser, and Akihiro Kanimori, and their team of ten contributors (Oliver Deiser, Jürgen Elstrodt, Ulrich Felgner, Michael Hallett, Albert Henrichs, Paul Larson, Charles Parsons, Gregory Taylor, Dirk van Dalen, and Dieter Wolke) are to be congratulated on their achievement.
Ebbinghaus, H.-D. Ernst Zermelo. An approach to his life and work. In cooperation with Volker Peckhaus. Springer, Berlin, 2007
Moore, Gregory H. Zermelo's axiom of choice. Its origins, development, and influence. Studies in the History of Mathematics and Physical Sciences, 8. Springer, New York, 1982.
Jeremy Gray is Professor of the History of Mathematics at the Open University, and an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. In 2009 he was awarded the Albert Leon Whiteman Memorial Prize by the American Mathematical Society for his work in the history of mathematics. His latest book is Plato’s Ghost: The Modernist Transformation of Mathematics.