Finsler Set Theory: Platonism and Circularity: Translation of Paul Finsler's papers on set theory with introductory comments by David Booth
Finsler's papers on set theory are presented, here for the first time in English translation, in three parts, 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 coinductive 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.
I. Philosophical Part.- (Renatus Ziegler).- Intrinsic Analysis of Antinomies and Self-Reference (Renatus Ziegler).- - Are there Contradictions in Mathematics? .- - Formal Proofs and Decidability [1926a].- - On the Solution of Paradoxes [1927b].- - Are there Undecidable Propositions? .- - The Platonistic Standpoint in Mathematics [1956a].- - Platonism After All [1956b].- II. Foundational Part.- (David Booth).- - On the Foundations of Set Theory, Part I [1926b].- - The Existence of the Natural Numbers and the Continuum .- - Concerning a Discussion on the Foundations of Mathematics [1941b].- - The Infinity of the Number Line .- - On the Foundations of Set Theory, Part II .- III. Combinatorial Part.- (David Booth).- The Combinatorics of Non-Well-Founded Sets (David Booth).- - Totally Finite Sets .- - On the Goldbach Conjecture .- Chronology of Paul Finsler’s Life.