Hausdorff's maximum chain sentence

The maximal chain theorem , also referred to as the maximality principle by Hausdorff , English Hausdorff's maximal principle , is a fundamental principle of both set theory and order theory . Felix Hausdorff published his maximality principle in 1914 in his important work Grundzüge der setlehre . The maximal chain theorem is closely connected with the lemma of Zorn and is logically equivalent to this and thus also (within the framework of set theory on the basis of the Zermelo-Fraenkel axioms ) to the axiom of choice .

formulation

The maximality principle can be formulated as follows:

Given is a partially ordered set and therein a subset that represents a chain with respect to the given order relation , i.e. That is, for every two elements and of either or applies ${\displaystyle (P,\leq )}$ ${\displaystyle K,}$ ${\displaystyle \leq }$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle K}$ ${\displaystyle k_{1}\leq k_{2}}$ ${\displaystyle k_{2}\leq k_{1}.}$

Then there is a comprehensive chain of which in turn is not included in any other chain of genuine . ${\displaystyle K}$ ${\displaystyle K_{0}}$ ${\displaystyle (P,\leq ),}$ ${\displaystyle (P,\leq )}$

In short, the maximality principle says that in an ordered set every chain can be extended to a maximal chain with regard to the inclusion relation . This also motivates the name of the principle as a maximum chain sentence.

Derivation from the axiom of choice according to Paul Halmos

An easily comprehensible direct derivation of the maximum chain theorem from the axiom of choice (without using the well-ordered theorem ) is given by Walter Rudin in the appendix of his well-known textbook Reelle und Complex Analysis. As Rudin shows, the decisive step in the proof lies in the following proposition, which Paul Halmos uses in his textbook Naive Set Theory (see literature ) to derive the lemma of Zorn from the axiom of choice .

Halmos' proposition

Let a given basic set and a non-empty inductive subset system in the associated power set be a subset system with the property that for every non-empty chain of subsets their union belongs in turn . ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}\subseteq 2^{X}}$ ${\displaystyle 2^{X},}$ ${\displaystyle {\mathcal {K}}\subseteq {\mathcal {F}}}$ ${\displaystyle \bigcup {\mathcal {K}}}$ ${\displaystyle {\mathcal {F}}}$

Let there be a function with for so that the following two properties are fulfilled: ${\displaystyle g\colon {\mathcal {F}}\to {\mathcal {F}}}$ ${\displaystyle A\mapsto g(A)}$ ${\displaystyle A\in {\mathcal {F}},}$

(1) ${\displaystyle A\subseteq g(A)}$

(2) ${\displaystyle |g(A)\setminus A|\leq 1}$

Then there is a with ${\displaystyle A_{1}\in {\mathcal {F}}}$ ${\displaystyle g(A_{1})\ =A_{1}.}$

Actual derivation

For the given partially ordered set, let the set system of the chains with respect to within ${\displaystyle (P,\leq )}$ ${\displaystyle {\mathcal {F}}\subseteq 2^{P}}$ ${\displaystyle \leq }$ ${\displaystyle P.}$

${\displaystyle {\mathcal {F}}}$is always non-empty and an inductive system of quantities.

The presupposed axiom of choice now ensures the existence of a selection function for one function with for all ${\displaystyle P,}$ ${\displaystyle f\colon 2^{P}\setminus \{\emptyset \}\to P}$ ${\displaystyle A\mapsto f(A)\in A}$ ${\displaystyle A\in 2^{P}\setminus \{\emptyset \}.}$

So you bet for ${\displaystyle A\in {\mathcal {F}}}$

${\displaystyle A^{*}=\{x\in {P\setminus A}\mid A\cup \{x\}\in {\mathcal {F}}\}}$

and then defines:

${\displaystyle g(A)={\begin{cases}A&{\text{für }}A^{*}=\emptyset \\{A\cup \{f(A^{*})\}}&{\text{für }}A^{*}\neq \emptyset \end{cases}}}$

According to Halmos' proposition, at least one ${\displaystyle A_{1}\in {\mathcal {F}}\colon }$

${\displaystyle {A_{1}}^{*}=\emptyset }$

According to the definition, this is a maximum element of in terms of the inclusion relation ${\displaystyle {A_{1}}^{*}}$ ${\displaystyle {\mathcal {F}}.}$

This conclusion shows that the axiom of choice entails Hausdorff's maximum chain theorem.

Historical notes

Felix Hausdorff published the maximum chain theorem in 1914 in his important work Grundzüge der setlehre . The formulation given above is that which is commonly used in the mathematical literature. Strictly proven - based on the well-ordered theorem -  Felix Hausdorff has an equivalent and only apparently weaker version in the main features :

In an ordered set there is always at least one chain that is not included in any other chain of real. ${\displaystyle (P,\leq )}$ ${\displaystyle (P,\leq )}$

Hausdorff points out in a comment following his proof that the maximum chain theorem in its formulation above can also be derived with a very similar proof.

Some authors of English-language literature assign the maximum chain sentence to Kazimierz Kuratowski and refer to it as the Kuratowski Lemma. With regard to the context of the history of mathematics, it should be noted that the maximum chain set was discovered or rediscovered several times in a different, but equivalent, form. The best-known example is probably the lemma of anger .

In this context, it is interesting to note that Walter Rudin in his real and complex analysis that the proof of the maximal chain theorem by way of the auxiliary theorem by Halmos is similar to the one that Ernst Zermelo presented in 1908 as the second derivation of the well-order theorem from the axiom of choice .

Moore's monograph gives a detailed account of the development history of the axiom of choice, the theorem of well-being, the maximum chain theorem, the Lemma of Zorn and other equivalent maximum principles (see literature ).

literature

Original work

• Ernst Zermelo : Proof that a lot can be well organized . In: Math. Ann. tape 59 , 1904, pp. 514-516 .
• Ernst Zermelo : New proof of the possibility of a well-ordered system . In: Math. Ann. tape 65 , 1908, pp. 107-128 .

Monographs

• E. Brieskorn , SD Chatterji et al. (Ed.): Felix Hausdorff. Collected Works . Volume II: Fundamentals of set theory. Springer-Verlag, Berlin (inter alia) 2002, ISBN 3-540-42224-2 , Chapter 6, § 1 ( books.google.de ).
• Oliver Deiser: Introduction to set theory . Springer-Verlag, Berlin (inter alia) 2002, ISBN 3-540-42948-4 .
• Keith Devlin : The Joy of Sets . 2nd Edition. Springer-Verlag, New York a. a. 1993, ISBN 0-387-94094-4 .
• Alan G. Hamilton: Numbers, sets and axioms. The apparatus of mathematics . Cambridge University Press, Cambridge 1982, ISBN 0-521-24509-5 .
• Paul Halmos : Naive set theory . Vandenhoeck & Ruprecht, Göttingen 1976, ISBN 3-525-40527-8 .
• Egbert Harzheim : Ordered Sets (=  Advances in Mathematics . Volume 7 ). Springer Verlag, New York 2005, ISBN 0-387-24219-8 , pp. 206 ff . ( MR2127991 ).
• Felix Hausdorff : Fundamentals of set theory . Chapter 6, § 1, Veit & Comp., Leipzig 1914 (reproduced in Srishti D. Chatterji et al. (Ed.): Felix Hausdorff. Collected works. Volume II: Fundamentals of set theory. Springer, Berlin 2002, ISBN 3-540-42224 -2 books.google.de ).
• John L. Kelley : General topology. Reprint of the 1955 edition published by Van Nostrand . Springer-Verlag, Berlin / Heidelberg / New York 1975, ISBN 3-540-90125-6 .
• Gregory H. Moore: Zermelo's axiom of choice . Springer-Verlag, Berlin / Heidelberg / New York 1982, ISBN 3-540-90670-3 .
• Walter Rudin : Real and Complex Analysis . 2nd Edition. Oldenbourg Wissenschaftsverlag, Berlin 2009, ISBN 978-3-486-59186-6 .

References and comments

1. ↑ ^{a b} Basics of set theory . S. 140-141 .
2. ↑ See for example Brieskorn, Chatterji et al.: Gesammelte Werke . tape II , 2002, p. 602-604 . and Harzheim: Ordered Sets . 2005, p. 50-52 .
3. ^ Walter Rudin : Real and Complex Analysis . 2nd Edition. Oldenbourg Wissenschaftsverlag, Berlin 2009, ISBN 978-3-486-59186-6 , p. 473-475, 483-484 .
4. ↑ The proof of this auxiliary theorem can be given within the framework of the Zermelo-Fraenkel set theory without using the axiom of choice .
5. ↑ chain in relation to the inclusion relation
6. ↑ Since Zorn's lemma can be deduced from the maximal chain theorem and this in turn implies the axiom of choice , one finds that there are three logically equivalent principles.
7. ↑ About Kelley or Hamilton; see literature !
8. ↑ See Brieskorn, Chatterji et al.: Collected works . tape II , p. 603 .
9. ↑ Zorn's lemma is therefore also referred to as the Kuratowski-Zorn lemma ; see. Brieskorn, Chatterji and others: Collected works . tape II , p. 603 .
10. ^ Walter Rudin : Real and Complex Analysis . 2nd Edition. Oldenbourg Wissenschaftsverlag, Berlin 2009, ISBN 978-3-486-59186-6 , p. 483-484 .