Abstract

Paul Finsler (1894-1970) had already secured renown as a differential geometer when he first took up set theory. His work in this field is heir to the spirit of the set theory put forward by Cantor who, as Finsler, was an uncompromising Platonist. Finsler's papers on set theory are presented, here for the first time in English translation, in three parts. Each reflects one of the three central concerns of his investigations, namely the philosophical, the foundational and the combinatorial approach, and each is preceded by an introduction to the field written by the editors. In the philosophical part of his work Finsler develops his approach to the paradoxes, his attitude toward formalized theories and his defense of Platonism in mathematics. He insisted on the existence of a conceptual realm within mathematics that transcends formal systems. From the foundational point of view, Finsler's set theory contains a strengthened criterion for set identity and a conductive specification of the universe of sets. The notion of the class of circle free sets introduced by Finsler is potentially a very fertile one although not very widespread today. Combinatorially, Finsler considers sets as generalized numbers to which one may apply arithmetical techniques. The introduction to this third section of the book extends Finsler's theory to non-well-founded sets. The present volume makes Finsler's papers on set theory accessible at long last to a wider group of mathematicians, philosophers and historians of science. A technical background is not necessary to appreciate the satisfying interplay of philosophical and mathematical ideas that characterizes this work.