**History of Mathematics**

Volume: 25;
2005;
322 pp;
Softcover

MSC: Primary 01; 03; 06; 26;

Print ISBN: 978-0-8218-3788-7

Product Code: HMATH/25

List Price: $76.00

Individual Member Price: $60.80

**Electronic ISBN: 978-1-4704-3893-7
Product Code: HMATH/25.E**

List Price: $76.00

Individual Member Price: $60.80

#### Supplemental Materials

# Hausdorff on Ordered Sets

Share this page *Edited by *
*J. M. Plotkin*

A co-publication of the AMS and the London Mathematical Society

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.

#### Table of Contents

# Table of Contents

## Hausdorff on Ordered Sets

- Cover Cover11
- Photo ii3
- Title page iii4
- Dedication v6
- Photo vi7
- Contents vii8
- Preface ix10
- Selected Hausdorff bibliography 120
- Introduction to "About a certain kind of ordered sets" 322
- About a certain kind of ordered sets [H 1901b] 1130
- Introduction to "The concept of power in set theory" 2342
- The concept of power in set theory [H 1904a] 3150
- Introduction to "Investigations into order types, I, II, III" 3554
- Investigations into order types [H 1906b] 4564
- Introduction to "Investigations into order types IV, V" 97116
- Investigations into order types [H 1907a] 113132
- Introduction to "About dense order types" 173192
- About dense order types [H 1907b] 175194
- Introduction to "The fundamentals of a theory of ordered sets" 181200
- The fundamentals of a theory of ordered sets [H 1908] 197216
- Introduction to "Graduation by final behavior" 259278
- Graduation by final behavior [H 1909a] 271290
- Appendix. Sums of ℵ₁ sets [H 1936b] 303322
- Bibliography 317336
- Back Cover Back Cover1343

#### 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