Hausdorff on Ordered Sets
About this Title
J. M. Plotkin, Michigan State University, East Lansing, MI, Editor
Publication: History of Mathematics
Publication Year: 2005; Volume 25
ISBNs: 978-0-8218-3788-7 (print); 978-1-4704-3893-7 (online)
MathSciNet review: MR2187098
MSC: Primary 01A75; Secondary 01A60, 03-03, 06-03, 26-03
Georg Cantor, the founder of set theory, published his last paper on sets in 1897. In 1900, David Hilbert made Cantor's Continuum Problem and the challenge of well-ordering the real numbers the first problem in his famous Paris lecture. It was time for the appearance of the second generation of Cantorians.
They emerged in the decade 1900–1909, and foremost among them were Ernst Zermelo and Felix Hausdorff. Zermelo isolated the Choice Principle, proved that every set could be well-ordered, and axiomatized the concept of set. He became the father of abstract set theory. Hausdorff eschewed foundations and pursued set theory as part of the mathematical arsenal. He was recognized as the era's leading Cantorian.
From 1901–1909, Hausdorff published seven articles in which he created a representation theory for ordered sets and investigated sets of real sequences partially ordered by eventual dominance, together with their maximally ordered subsets. These papers are translated and appear in this volume. Each is accompanied by an introductory essay. These highly accessible works are of historical significance, not only for set theory, but also for model theory, analysis and algebra.
Graduate students and researchers interested in set theory and the history of mathematics.
Table of Contents
- Selected Hausdorff bibliography
- Introduction to "About a certain kind of ordered sets"
- About a certain kind of ordered sets [H 1901b]
- Introduction to "The concept of power in set theory"
- The concept of power in set theory [H 1904a]
- Introduction to "Investigations into order types, I, II, III"
- Investigations into order types [H 1906b]
- Introduction to "Investigations into order types IV, V"
- Investigations into order types [H 1907a]
- Introduction to "About dense order types"
- About dense order types [H 1907b]
- Introduction to "The fundamentals of a theory of ordered sets"
- The fundamentals of a theory of ordered sets [H 1908]
- Introduction to "Graduation by final behavior"
- Graduation by final behavior [H 1909a]
- Appendix. Sums of $\aleph _1$ sets [H 1936b]