You are here

Hausdorff on Ordered Sets

J. M. Plotkin, editor
American Mathematical Society and London Mathematical Society
Publication Date: 
Number of Pages: 
History of Mathematics Sources 25
[Reviewed by
Paul E. Cohen
, on

Between 1901 and 1936, Felix Hausdorff published a series of papers on the general subject of the orderings of sets of real numbers.  J.M. Plotkin has made these papers accessible to the many mathematicians today who have difficulty reading the German language. 

In all, this book includes the translations of seven papers.  Merely translating these papers into English would have been a great service to the mathematical community, but even in English translation these papers remain difficult reading.  In addition to the usual difficulty of reading tightly constructed mathematical writing, there is the problem of the historical nature of these papers.  Much of Hausdorff's notation and some of his terminology now seem archaic.

To accompany each of the translated papers, Plotkin has written substantial notes.  These notes include historical information along with helpful explanations of the notation and concepts.  The reader should be prepared to flip back and forth between Hausdorff's papers and their accompanying notes because the notes so admirably support the reading of the papers.

Hausdorff died in 1942, and one might conclude that this book is primarily of historical interest.  It certainly is a valuable book for students of the history of mathematics, but this does not mean it should be ignored by mathematics researchers.  Before his death in 1978, Gödel was actively encouraging young researchers at the Institute for Advanced Study to pay more attention to these very papers by Hausdorff. 

In 1938, Gödel proved the consistency of the continuum hypothesis (CH) with the Zermelo-Frankel axioms (ZF) and in 1963, Paul J. Cohen proved CH to be independent of ZF.  It is tempting to conclude that CH is now thoroughly settled mathematics.  But the very same statements (even the attributions and dates) can be made about the consistency and independence of the axiom of choice (AC) with ZF.  And yet AC is settled mathematics in a much stronger sense than is CH.  Namely, there is a broad consensus in the mathematical community that AC is true; the conventional axiom system for set theory is ZFC, not just ZF.  In contrast, CH remains in a quite undetermined state with respect to its truth.

It is certainly unsettling to a Platonist to have a concept that is as intuitively meaningful as is the CH remain in limbo, condemned forever to remain neither true nor false.  Gödel's hope was that some expanded study of Hausdorff's work would eventually lead to a consensus about the truth of CH.

On page 114 of Plotkin's book is the statement of a lemma that is at the core of this idea:

An everywhere dense type without ww* limits and ww* gaps ... if it also has no WW*-gaps, then its cardinality is > À 1

and in particular, if the ordering is of a set of real numbers then CH cannot be true. 

It is probably not entirely a justified conclusion, but it is not difficult to convince oneself that this lemma is the organizing principle behind these papers of Hausdorff and that this lemma is also the motivation for collecting these papers together into this book.  It certainly does seem to be the concept that links the study of these order types with the issue of the continuum hypothesis.

Paul Cohen received his Ph.D. from the University of Illinois, was appointed as a Member of the Institute for Advanced Study by Kurt Gödel, and has taught at the University of Tennessee and at Lehigh University. He currently lives in Maine and is teaching at Colby College.

  • 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]
  • Bibliography