Teichmüller–Tukey lemma

The lemma of Teichmüller - Tukey (after Oswald Teichmüller and John W. Tukey ), sometimes just called Tukey 's Lemma, is a set of set theory. It is in the context of set theory based on the ZF axioms equivalent to the axiom of choice and thus also to the Lemma of Zorn, the Hausdorff maximal chain set and the well-ordering theorem.

To formulate the statement we need the notion of the finite nature of a quantity. A lot has finite character if

It follows easily that for every all subsets ( not only finite) are elements of: .

There are two different formulations of the lemma:

• If a non-empty set of finite character, is imposed on the set inclusion a maximal element.
• If a non-empty set of finite character and is, so there is with respect to the set inclusion with a maximal element.