Talk:Hausdorff maximal principle

Maximal

When is an totally ordered subset a maximal totally ordered subset? Is that when you cannot add another element without breaking total-orderedness? -- Jan Hidders 20:27 Sep 8, 2002 (UTC)

I'm guessing, but that sounds right. I would put it that no element can be added and still preserve total-orderedness. Same thing though. -- Tarquin 20:32 Sep 8, 2002 (UTC)

Yes, that's right. Note that it's not a maximally totally ordered subset (what would that even mean?), but a totally ordered subset which is maximal among the totally ordered subsets. We should probably clarify this.AxelBoldt 23:06 Sep 8, 2002 (UTC)

Obvious

Why an equivalent form of the theorem is that in every partially ordered set there exists a maximal totally ordered subset is not obvious? Just start with a one element set, and extend it. Albmont 21:07, 9 December 2006 (UTC)

After you get to infinity, then what? What if, as in general topology, your partially ordered set contains an uncountable infinity of elements? linas 01:20, 8 April 2007 (UTC)

Order of the article

In the article a formulation of the Hausdorff maximality principle is given, followed by an equivalent statement of it. What follows is a formal statement. To my surprise, this is a formal statement of the equivalent form of the principle. I myself find this a bit weird. Wouldn't it be better if the formal statement would correspond to the first statement of the principle? —Preceding unsigned comment added by HSNie (talk • contribs) 19:46, 29 May 2009 (UTC)

I've reformulated it. However, I am unconvinced of the usefulness of the whole section. It's not as if the statement there is any more formal that the short one given in the lead, it's only slightly more explicit by expanding some of the definitions. — Emil J. 13:08, 1 June 2009 (UTC)

Neither am I. It isn't a formal formulation at all. I just added the prove that the equivalent form is equivalent in a form that isn't complete yet. If someone feels like completing the details I just omitted, it'd be great. Hope you agree with adding it.HSNie 23:21, 3 June 2009 (UTC) —Preceding unsigned comment added by HSNie (talk • contribs)

"Proof"?

The statement in the first sentence that Hausdorff "proved" it in 1914 needs clarification: he can't really have proved the statement in an absolute sense as it doesn't follow from ZFC. I guess it means he proved it to be a consequence of the axiom of choice, or to be equivalent to the axiom of choice?! — Preceding unsigned comment added by 130.88.0.182 (talk) 17:23, 24 January 2017 (UTC)

In Kapitel (Chapter) 1, §14 of Grundzüge der Mengenlehre (page 39), Hausdorff emphasizes that the principle is not a theorem derived from earlier axioms but an independent assumption about sets, essentially another axiom. Unlike Zermelo, whose earlier axiomatization of set theory was formal, Hausdorff was aiming for the sort of less formal account of sets that Paul Halmos later called naive set theory. He nevertheless used Zermelo's axioms as guidance to avoid the paradoxes of set theory. His later book Mengenlehre acknowledged the main equivalent statements of the Axiom of Choice and didn't even bother to mention that his maximality principle was one of them, perhaps because it was so close to Zorn's lemma. Vaughan Pratt (talk) 05:57, 9 April 2026 (UTC)

Vaughan Pratt (talk) 05:57, 9 April 2026 (UTC)

Examples from Munkres

The two examples on this page are copied verbatim from Munkres, Chapter 1, Section 11. — Preceding unsigned comment added by 2.31.167.131 (talk) 11:17, 13 July 2018 (UTC)

Links to Planetmath

The Planetmath reference template seems to be broken. Here are the direct links.

• Hausdorff's maximum principle
• A proof of equivalence of Zorn's lemma, the well-ordering theorem, and Hausdorff's maximum principle

-- Lilalas (talk) 15:18, 28 February 2019 (UTC)