Zorn's lemma

We show that Zorn's lemma is a consequence of the axiom of choice.

Theorem A.1.1Assume the axiom of choice. Suppose that (X, ≤) is a non-empty partially ordered set with the property that every non-empty chain (totally ordered subset) of X has an upper bound. Then there exists a maximal element in X.

Proof We need a few more definitions. Suppose that A is a subset of a partially ordered set (X, ≤) and that x ∈ X. x is a strict upper bound for A if a < x for all a ∈ A. A totally ordered set (S, ≤)is well-ordered if every non-empty subset of S has a least element. A subset D of a totally ordered set (S, ≤)is an initial segment of S if whenever x ∈ S, d ∈ D and x ≤ d then x ∈ D.

We break the proof into a sequence of lemmas and corollaries. If A is a subset of X, let A be the set of strict upper bounds of A. If A = ∅, then A = X.

Let C be the set of chains in X.

Lemma A.1.2Suppose that there exists C ∈C for which C = ∅. Then C has a unique upper bound, which is a maximal element of X.