Tukey's Lemma

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Theorem

Formulation 1

Let $S$ be a non-empty set of finite character.

Then $S$ has an element which is maximal with respect to the subset relation.


Formulation 2

Let $S$ be a non-empty set of finite character.

Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.


Axiom of Choice

This theorem depends on the Axiom of Choice, by way of Zorn's Lemma.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.


Also known as

Tukey's Lemma is still occasionally found with the name of Teichmüller attached to it, but this is dying out.


Also see

  • Results about Tukey's lemma can be found here.


Source of Name

This entry was named for John Wilder Tukey.