Maximal Principles are Equivalent

Theorem

The Maximal Principles are equivalent:

• Kuratowski's Lemma

• Tukey's Lemma

• Zorn's Lemma

• Hausdorff's Maximal Principle

Proof

We have:

• Kuratowski's Lemma implies Tukey's Lemma

• Tukey's Lemma implies Zorn's Lemma

• Zorn's Lemma implies Hausdorff's Maximal Principle

• Hausdorff's Maximal Principle implies 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: $\S 5$ Maximal principles: Proposition $5.11$