Editing Felix Hausdorff (section)

===Theory of ordered sets===
Hausdorff's entrance into a thorough study of ordered sets was prompted in part by Cantor's continuum problem: where should the [[cardinal number]] <math>\aleph = 2^{\aleph_0}</math> be placed in the sequence <math>\{\aleph_{\alpha}\}</math>? In a letter to Hilbert on 29 September 1904, he speaks of this problem, "it has plagued me almost like [[monomania]]".<ref>Niedersächsische Staats- und Universitätsbibliothek zu Göttingen, Handschriftenabteilung, NL Hilbert, Nr. 136.</ref> Hausdorff saw a new strategy to attack the problem in the set <math> \mathrm{card} (T(\aleph_0)) = \aleph</math>. Cantor had suspected <math>\aleph = \aleph_1</math>, but had only been able to show that <math>\aleph \geq \aleph_1</math>. While <math>\aleph_1</math> is the "number" of possible [[well-ordering]]s of a [[countable set]], <math>\aleph</math> had now emerged as the "number" of all possible orders of such an amount. It was natural, therefore, to study systems that are more specific than orders, but more general than well-orderings. Hausdorff did just that in his first volume of 1901, with the publication of theoretical studies of "graded sets". However, we know from the results of [[Kurt Gödel]] and [[Paul Cohen (mathematician)|Paul Cohen]] that this strategy to solve the continuum problem is just as ineffectual as Cantor's strategy, which was aimed at generalizing the [[Cantor&ndash;Bendixson theorem|Cantor–Bendixson principle]] from [[closed set]]s to general uncountable sets.

In 1904 Hausdorff published the recursion named after him, which states that for each non-limit ordinal <math>\mu</math> we have <math>\aleph_{\mu}^{\aleph_{\alpha}} = \aleph_{\mu} \; \aleph_{\mu -1}^{\aleph_{\alpha}}.</math>

This formula was, together with a later notion called cofinality introduced by Hausdorff, the basis for all further results for [[Aleph exponentiation]]. Hausdorff's excellent knowledge of recurrence formulas of this kind also empowered him to uncover an error in [[Julius König]]'s lecture at the [[International Congress of Mathematicians]] in 1904 in [[Heidelberg]]. There König had argued that the continuum cannot be well-ordered, so its cardinality is not an Aleph at all, and thus caused a great stir. The fact that it was Hausdorff who clarified the mistake carries a special significance, since a false impression of the events in Heidelberg lasted for over 50 years.<ref>Detaillierte Angaben findet man in den gesammelten Werken, Band&nbsp;II, S.&nbsp;9–12.</ref>

In the years 1906–1909 Hausdorff did his groundbreaking and fundamental work on ordered sets. Of fundamental importance to the whole theory is the concept of [[cofinal (mathematics)|cofinal]]ity, which Hausdorff introduced.  If <math>\alpha</math> is an ordinal, then a collection <math>\mathcal{A}</math> of ordinals each less than <math>\alpha</math> is cofinal in <math>\alpha</math> if, for every <math>\beta<\alpha</math>, there is a <math>\gamma\in\mathcal{A}</math> such that <math>\beta\leq\gamma</math>. So, in a sense, a cofinal collection in <math>\alpha</math> encompasses all of <math>\alpha</math>. Since a collection of objects from a well-ordered set is well-ordered under the original ordering, we can naturally assign a well-order to any such <math>\mathcal{A}</math>. As it turns out, any collection <math>\mathcal{A}</math> of ordinals less than an ordinal <math>\alpha</math> has an [[order type]] no greater than <math>\alpha</math>. An ordinal <math>\alpha</math> is then called regular if any collection of ordinals cofinal in <math>\alpha</math> has, in fact, the same order type as <math>\alpha</math>.  In other words, <math>\alpha</math> is regular if every collection of smaller ordinals that encompasses <math>\alpha</math> has the same size, order-theoretically, as <math>\alpha</math>. Hausdorff's question, whether there are regular numbers which index a limit ordinal, was the starting point for the theory of [[inaccessible cardinals]]. Hausdorff had already noticed that such numbers, if they exist, must be of "exorbitant size".<ref>H.: Gesammelte Werke. Band&nbsp;II: ''Grundzüge der Mengenlehre.'' Springer-Verlag, Berlin, Heidelberg etc. 2002. Kommentare von U.&nbsp;Felgner, S.&nbsp;598–601.</ref>

The following theorem due to Hausdorff is also of fundamental importance: for each unbounded and ordered dense set <math>A</math> there are two uniquely determined regular initial numbers <math>\omega_{\xi}, \omega_{\eta}</math> so that <math>A</math> is cofinal with <math>\omega_{\xi}</math> and coinitial with <math>\omega_{\eta}^*</math> (where * denotes the inverse order). This theorem provides, for example, a technique to characterize elements and gaps in ordered sets.

If <math>W</math> is a predetermined set of characters (element and gap characters), the question arises whether there are ordered sets whose character set is exactly <math>W</math>. One can easily find a necessary condition for <math>W</math>, but Hausdorff was also able to show that this condition is sufficient. For this one needs a rich reservoir of ordered sets, which Hausdorff was also able to create with his theory of general products and powers.<ref>H.: Gesammelte Werke. Band&nbsp;II: ''Grundzüge der Mengenlehre.'' Springer-Verlag, Berlin, Heidelberg etc. 2002. S.&nbsp;604–605.</ref> In this reservoir can be found interesting structures like the Hausdorff <math>\eta_{\alpha}</math> normal-types, in connection with which Hausdorff first formulated the [[generalized continuum hypothesis]]. Hausdorff's <math>\eta_{\alpha}</math>-sets formed the starting point for the study of the important model theory of [[saturated structure]].<ref>Siehe dazu den Essay von U. Felgner: ''Die Hausdorffsche Theorie der <math>\eta_{\alpha}</math>-Mengen und ihre Wirkungsgeschichte'' in H.: Gesammelte Werke. Band&nbsp;II: ''Grundzüge der Mengenlehre''. Springer-Verlag, Berlin, Heidelberg etc. 2002. S.&nbsp;645–674.</ref>

Hausdorff's general products and powers of cardinalities led him to study the concept of partially ordered set. The question of whether any ordered subset of a partially ordered set is contained in a maximal ordered subset was answered in the positive by Hausdorff using the well-ordering theorem. This is the [[Hausdorff maximal principle]], which follows from either the well-ordering theorem or the axiom of choice, and as it turned out, is also equivalent to the axiom of choice.<ref>Siehe dazu und zu ähnlichen Sätzen von [[Kazimierz Kuratowski|Kuratowski]] und [[Max August Zorn|Zorn]] den Kommentar von U.&nbsp;Felgner in den gesammelten Werken, Band&nbsp;II, S.&nbsp;602–604.</ref>

Writing in 1908, [[Arthur Moritz Schoenflies]] found in his report on set theory that the newer theory of ordered sets (i.e., that which occurred after Cantor's extensions) was almost exclusively due to Hausdorff.<ref>Schoenflies, A.: ''Die Entwickelung der Lehre von den Punktmannigfaltigkeiten.'' Teil&nbsp;II. Jahresbericht der DMV, 2.&nbsp;Ergänzungsband, Teubner, Leipzig 1908., S.&nbsp;40.</ref>