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Hausdorff on Ordered Sets
Edited by: J. M. Plotkin, Michigan State University, East Lansing, MI
A co-publication of the AMS and the London Mathematical Society.
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History of Mathematics
2005; 322 pp; softcover
Volume: 25
ISBN-10: 0-8218-3788-5
ISBN-13: 978-0-8218-3788-7
List Price: US$76
Member Price: US$60.80
Order Code: HMATH/25
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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 of his famous lecture at the International Congress in Paris. Thus, as the nineteenth century came to a close and the twentieth century began, Cantor's work was finally receiving its due and Hilbert had made one of Cantor's most important conjectures his number one problem. It was time for the second generation of Cantorians to emerge.

Foremost among this group 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 developed set theory as a branch of mathematics worthy of study in its own right, capable of supporting both general topology and measure theory. He is recognized as the era's leading Cantorian.

Hausdorff published seven articles in set theory during the period 1901-1909, mostly about ordered sets. This volume contains translations of these papers with accompanying introductory essays. They are highly accessible, historically significant works, important not only for set theory, but also for model theory, analysis and algebra.

This book is suitable for graduate students and researchers interested in set theory and the history of mathematics.

This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought.

Also available from the AMS by Felix Hausdorff are the classic works, Grundzüge der Mengenlehre (Volume 61) and Set Theory (Volume 119), in the AMS Chelsea Publishing series.

Readership

Graduate students and researchers interested in set theory and the history of mathematics.

Reviews

"...this volume greatly facilitates the access of the international readership to Hausdorff's early contributions to set theory and gives detailed information on the history of set theory in general. It is a very welcome, probably even necessary, complement to the ongoing enterprise of editing Hausdorff's Gesammelta Werke. It will be of great help to anybody interested in the historical and mathematical development of 20th-century set theory and logic."

-- Historia Mathematica

Table of Contents

  • J. M. Plotkin -- Selected Hausdorff bibliography
  • J. M. Plotkin -- Introduction to "About a certain kind of ordered sets"
  • F. Hausdorff -- About a certain kind of ordered sets [H 1901b]
  • J. M. Plotkin -- Introduction to "The concept of power in set theory"
  • F. Hausdorff -- The concept of power in set theory [H 1904a]
  • J. M. Plotkin -- Introduction to "Investigations into order types, I, II, III"
  • F. Hausdorff -- Investigations into order types [H 1906b]
  • J. M. Plotkin -- Introduction to "Investigations into order types IV, V"
  • F. Hausdorff -- Investigations into order types [H 1907a]
  • J. M. Plotkin -- Introduction to "About dense order types"
  • F. Hausdorff -- About dense order types [H 1907b]
  • J. M. Plotkin -- Introduction to "The fundamentals of a theory of ordered sets"
  • F. Hausdorff -- The fundamentals of a theory of ordered sets [H 1908]
  • J. M. Plotkin -- Introduction to "Graduation by final behavior"
  • F. Hausdorff -- Graduation by final behavior [H 1909a]
  • F. Hausdorff -- Appendix. Sums of \(\aleph_1\) sets [H 1936b]
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