The trio: Choice = Zorn = Well-ordering
Parent: order-theory
Equivalent over ZFAxiom of Choice ⟺ Zorn's lemma ⟺ Well-ordering theorem. All three are equivalent given the ZF axioms — and none is provable or refutable from ZF alone (Gödel + Cohen).
The three statements:
- Axiom of Choice (AC): every family of nonempty sets has a choice function (pick one element from each, all at once).
- Zorn's lemma: every poset in which every chain has an upper bound has a maximal element.
- Well-ordering theorem: every set can be well-ordered (given a total order with no infinite descent).
The cycle (each implies the next):
AC → Zorn (build a maximal chain by transfinite choice); Zorn → Well-ordering (a maximal partial well-ordering of the set must cover everything); Well-ordering → AC (well-order each set, then "choose the least" is a definable choice function). So the loop closes — they're one principle in three costumes.
"well-ordering" ≠ "well-founded"The trio member is the well-ordering theorem (you can impose a total well-order on any set), not the property well-founded. A well-order is a total well-founded order — but "every set is well-founded" isn't the claim; "every set can be well-ordered" is.
Why each costume is handy: Zorn for existence of maximal objects (a basis of any vector space, maximal ideals, algebraic closures); Well-ordering for transfinite recursion/induction over an arbitrary set; AC for plain choice functions.
The catch (non-constructive): it asserts objects with no recipe — hence the Banach–Tarski paradox (a ball cut into 5 pieces reassembled into two balls). Most mathematicians accept AC; constructivists reject it; and the Kleene/Knaster–Tarski fixed points and Bourbaki–Witt are the choice-free cousins.