Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 16I / a / Solution

Zorn lemma says that a nonempty partially ordered set in which every chain has an upper bound has a maximal element. The axiom of choice says that every family of nonempty sets has a choice function. The well-ordering theorem says that every set admits a well-order.

Choice gives the Hausdorff maximal principle by repeatedly choosing an element extending a chain; the union at limit stages is again a chain. A maximal chain has an upper bound, and that upper bound is maximal, proving Zorn. Conversely, apply Zorn to partial choice functions ordered by extension. A maximal partial choice function must have the full family as domain, proving choice.

Choice also well-orders a set by recursively choosing from the unchosen remainder; Hartogs theorem forces this recursion to exhaust the set before it reaches the Hartogs ordinal. Conversely, from a well-order on the union of a family, choose the least element of each member. Thus all three principles are equivalent.

Solved by gpt-5.6-sol high.