Zorn's lemma
Statement
Let be a non-empty partially ordered set in which every chain (totally ordered subset) has an upper bound in . Then has at least one maximal element.
Why is it true?
To find a maximal element, keep climbing to strictly larger elements to build a chain; the hypothesis guarantees the chain always has somewhere to 'cap off', so the climbing process cannot go on forever without ever reaching a top — a maximal element must exist.
Proof sketch
Assuming the axiom of choice, fix a choice function and build a chain by transfinite recursion: at each stage, if the current chain has a strict upper bound not yet in the chain, add one (chosen via the choice function); the hypothesis guarantees a bound exists at every limit stage too. This process cannot run past all ordinals (that would give a chain longer than any set can be), so it must stop at some maximal element. Conversely, Zorn's lemma applied to the poset of partial choice functions, ordered by extension, yields the axiom of choice — the two are equivalent over ZF.
Topics that use this theorem
Related theorems
Step-by-step proofs
No step-by-step proof yet for this theorem.
References
- Max Zorn (1935). A remark on method in transfinite algebra
- Kenneth Kunen (1980). Set Theory: An Introduction to Independence Proofs