The present paper is Zermelo's first set-theoretic publication. It appeared in the Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen in 1901. The addendum to the title reads: "Vorgelegt von D. Hilbert in der Sitzung vom 9. März 1901". In the period between 1894 and 1897 Zermelo was scientific assistant to Max Planck in Berlin. In 1897 he moved to Göttingen and wrote his Habilitationsschrift in hydrodynamics. Influenced by David Hilbert, he became gradually involved in set-theoretic questions. In the winter semester 1900/01 Zermelo gave a lecture on set theory—one of the first lectures at all on this then rather new subject. We may date the main result of the paper to late winter 1900/01.