Hausdorff on Ordered Sets

This book is devoted to seven of Hausdorff’s articles on Cantorian set theory (mostly about ordered sets) that were published during the period 1900 – 1910. Each article, translated by the author, is preceded by an essay that offers an illuminating commentary including references on the relationship to the work of others. Furthermore, a translation of Hausdorff’s article, “Sums 1א Sets”, from 1936 is in the appendix; it belongs to the circle of problems in question. Moreover, we can find in the preface a short outline of the substantial historical period of the birth and initial development of set-theoretically based modern mathematics. Recall that in the years 1900-1910 it was mostly E. Zermelo and F. Hausdorff who traced out two basic trends in the still new field of set theory. While E. Zermelo can be seen as the father of abstract set theory (remember his list of axioms of formal set theory) F. Hausdorff developed set theory as a useful tool and background to a study of concrete mathematical problems. In this spirit, he influenced both measure theory and the foundations of general topology. Thus the book gives material concerning fundamental knowledge of several branches of modern mathematics and it can be recommended as an illuminating description of the field, both historically and mathematically.

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