Lemma from Teichmüller-Tukey

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

To formulate the statement we need the concept of the finite character of a set. A lot has finite character , though ${\displaystyle {\mathcal {F}}}$

${\displaystyle Y\in {\mathcal {F}}\Leftrightarrow \forall \ Z{\text{ endlich }}\wedge Z\subseteq Y:Z\in {\mathcal {F}}}$.

It follows easily that for every all subsets (not just the finite) elements : are . ${\displaystyle Y\in {\mathcal {F}}}$ ${\displaystyle X\subseteq Y}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X\in {\mathcal {F}}}$

There are two different formulations of the lemma:

• If a non-empty set is of finite character, there is a maximum element with respect to the set inclusion. ${\displaystyle {\mathcal {F}}}$
• If a non-empty set is of finite character and is , then there is a maximal element with with respect to the set inclusion . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle A\in {\mathcal {F}}}$ ${\displaystyle B\in {\mathcal {F}}}$ ${\displaystyle A\subseteq B}$

Web links

• Lemma from Tukey on PlanetMath