Hausdorff maximal principle

The maximum chain set, also called maximality principle Hausdorff, is a fundamental principle of set theory. Felix Hausdorff published his maximality principle in 1914 in his important work basics of set theory. The maximum chain set is closely connected to the Lemma of Zorn and for this and thus also to (in the context of set theory based on the Zermelo -Fraenkel axioms ) axiom of choice logically equivalent.

• 4.1 Original Articles
• 4.2 monographs

Formulation

The maximality principle can be formulated as follows:

In brief, the maximality principle states so that in an ordered set, each chain can be extended to a maximum with respect to the inclusion relation chain. This motivates the name of the principle as a maximum chain set.

Derivation from the axiom of choice by Paul Halmos

A well comprehensible direct derivation of the maximum chain set from the axiom of choice (without use of the well-ordering theorem ) are Walter Rudin in the appendix to his famous textbook, Real and Complex Analysis. As Rudin shows the decisive step proof lies in the following lemma which is used by Paul Halmos in his textbook Naive set theory (see references ) to derive the lemma of anger from the axiom of choice.

Lemma of Halmos

Actual derivation

For the given partially ordered set the amount of system of chains is respect within

Is always non- empty and an inductive system of sets.

The presupposed axiom of choice now ensures the existence of a selection function that is a function with for all

This one is for

And then defining:

After Halmosschen lemma is now a for at least

This is now by definition with respect to the inclusion relation maximal element of

This conclusion shows that the axiom of choice draws the Hausdorff maximal chain set by itself.

Historical Notes

Felix Hausdorff published the maximum chain set in 1914, in his important work basics of set theory. The reproduced above formulation is the one that is in the mathematical literature usually called. Rigorously proven - starting from the well-ordering theorem - Felix Hausdorff has in the Broad equivalent and only apparently weaker version:

Hausdorff has, in a statement in response to his proof to the fact that the maximum chain kit can also be derived in its above formulation with a very similar proof.

Some authors of English literature assign the maximum chain set Kazimierz Kuratowski and call him a Kuratowski lemma. In terms of mathematics historical context it should be noted that the maximum chain set was repeatedly discovered or rediscovered in each other, but equivalent, form. The most prominent example here is probably the Lemma of Zorn.

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

On the Development of the axiom of choice, well-ordering theorem, maximum chain set, Lemma of anger and other equivalent maximum principles, the monograph by Moore gives a detailed representation ( see literature).