Parent: order-theory
Zorn's lemmaIn a partially ordered set where every chain (totally-ordered subset) has an upper bound, there exists a maximal element.
The equivalence (over ZF)Zorn's lemma ⟺ the Axiom of Choice ⟺ the Well-ordering theorem (every set can be well-ordered).
How it's used
The recipe for a non-constructive existence proof:
- Form the poset of partial solutions, ordered by extension.
- Show any chain has an upper bound — usually the union of the chain is itself a partial solution.
- Zorn gives a maximal partial solution.
- Argue a maximal one must be complete/total (else you could extend it, contradicting maximality).
Classic uses: every vector space has a basis; every proper ideal sits inside a maximal ideal; every field has an algebraic closure; Hahn–Banach; ultrafilters exist; Tychonoff (arbitrary products of compact spaces are compact).
Not a constructive fixed point
Distinguish from the choice-free order theorems in the neighborhood: Knaster–Tarski (complete lattice), Kleene (CPO, climb from ⊥), and Bourbaki–Witt (chain-complete + inflationary map) all give fixed/maximal points without choice. Zorn is the one that needs AC — it's the non-constructive existence tool. (Bourbaki–Witt is often the lemma used to prove Zorn.)