Hausdorff on Ordered Sets
American Mathematical Soc. - 322 pàgines
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. Also available from the AMS by Felix Hausdorff are the classic work, Grundzuge der Mengenlehre, and its English translation, Set Theory, as Volume 69 and Volume 119 in the AMS Chelsea Publishing series. Information for our distributors: Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.
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About a Certain Kind of Ordered Sets H 1901b ll
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
The Fundamentals of a Theory of Ordered Sets H 1908
Introduction to Graduation by Final Behavior
Appendix Sums of N Sets H 1936b
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aleph arbitrary argument base belong Bernstein Bois-Reymond boundary elements called Cantor's cofinal coinitial construction contains continued fraction Continuum Hypothesis convergent sequence corresponding countable set covering set decomposition defined denote dense set densely ordered differences divergent sequences end segment everywhere dense type exclusive intervals exists Felix Hausdorff finite follows functions fundamental sequences gaps graded set Hausdorff homogeneous types hypothesis initial piece initial segment inverse irreducible irreducible sets isomeric type J?-sequence last element limit number linear continuum middle segment monotonic numerical sequences order types ordered set ordinal number pantachie types powers principal element principal interval proof properties proved rational ordered domain real numbers regular initial number regular roots second infinite cardinality second number class secondary elements set of type set theory similar species subset transfinite transfinite induction types of cardinality u;*-sequence uncountable set well-ordered set Well-Ordering Theorem Zermelo