In the following exercise, we ask you to prove the equivalence of the choice axiom, the we...

In the following exercise, we ask you to prove the equivalence of the choice axiom, the well-ordering theorem, and the maximum principle. We comment that of these exercises, only Exercise 7 uses the choice axiom.

Theorem (General principle of recursive definition). Let J be a well-ordered set; let C be a set. Let  be the set of all functions mapping sections of J into C. Given a function ρ : → C, there exists a unique function h : J → C such that h (α) = ρ(h\Sα) for each α ∈J.

[Hint: Follow the pattern outlined in Exercise 10 of §10.]