Zorn's Lemma

Set Theory · Axiom Academy

The algebraist's favorite form of the Axiom of Choice Zorn's Lemma: Let (P, ≤) be a non-empty partially ordered set. If every chain in P has an upper bound in P, then P contains at least one maximal element. Chain: A subset C ⊆ P where any two elements are comparable (totally ordered) Upper bound: An element u ∈ P such that c ≤ u for all c ∈ C Maximal element: An element m ∈ P with no element strictly greater (if m ≤ x then m = x) 2. Understanding the Key Terms Chain: Think of a chain as a "linearly ordered" subset—any two elements can be compared. For example, in the poset of subsets under inclusion, a ⊆ a,b ⊆ a,b,c forms a chain. Upper bound: An element that is "at least as large" as everything in the chain. The upper bound need not be in the chain itself! Maximal element: An element with "nowhere to go up"—not necessarily the largest element, just one that can't be strictly exceeded. AC ⇒ Zorn: Given a poset where every chain has an upper bound, use AC to construct a transfinite sequence by repeatedly choosing elements above what we've built. This process must terminate at a maximal element. Zorn ⇒ AC: Given a collection of sets A i , consider the poset of partial choice functions ordered by extension. Chains have upper bounds (their unions), so by Zorn, there's a maximal partial choice function—which must be total!

This is the written version of the interactive lesson above. See the full Set Theory course.