Principle E is Equivalent to Kuratowski's Lemma

Theorem

Kuratowski's Lemma is equivalent to Principle E.

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 Principle E:

Let $S$ be a set of sets.

Let $S$ be closed under chain unions.

Then every element $b$ of $S$ is a subset of some element of $S$ that has no immediate extension in $S$.

$\Box$

Kuratowski's Lemma implies Principle E

Let Kuratowski's Lemma hold.

Let $x$ be a maximal element under the subset relation.

Then $x$ has no immediate extension which is not $x$ itself.

That is, Principle E holds directly.

$\Box$

Principle E implies Kuratowski's Lemma

Let Principle E hold.

Let $S$ be swelled.

That is, every subset of every element of $S$ is an element of $S$.

Then an element $x$ of $S$ has no immediate extension in $S$ if and only if $x$ is a maximal element of $S$ under the subset relation.

Thus Principle E implies that for every swelled set $S$ which is closed under chain unions, every $b \in S$ is a subset of some maximal element of $S$.

By Class of Finite Character is Swelled, a set of finite character is swelled.

By Class of Finite Character is Closed under Chain Unions a set of finite character is closed under chain unions.

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Thus Principle E implies Tukey's Lemma.

From Maximal Principles are Equivalent, we have that Tukey's Lemma is equivalent to Kuratowski's Lemma.

$\blacksquare$

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: $\S 6$ Another approach to maximal principles: Theorem $6.1$