Kuratowski's Lemma implies Tukey's Lemma

Theorem

Let Kuratowski's Lemma be accepted as true.

Then Tukey's Lemma holds.

Proof

Recall Kuratowski's Lemma:

Let $S$ be a set of sets which is closed under chain unions.

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

$\Box$

Recall Tukey's Lemma:

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.

$\Box$

So, let us assume Kuratowski's Lemma.

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

From Class of Finite Character is Closed under Chain Unions, $S$ is closed under chain unions.

Then by Kuratowski's Lemma:

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

Thus it is seen that Tukey's Lemma likewise holds.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles: Proposition $5.5$