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.

Let J and E be well-ordered sets; suppose there is an order-preserving map k : J → E. Using Exercises 1 and 2, show that J has the order type of E or a section of E. [Hint: Choose e0 ∈ E. Define h : J → E by the recursion formula

and h(α) = e0 otherwise. Show that h(α)<k(α) for all α; conclude that h(Sα) ≠ E for all α.]