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.
(a) Let J and E be well-ordered sets; let h : J → E. Show the following two statements are equivalent:
(i) h is order preserving and its image is E or a section of E.
(ii) h(α) = smallest [E − h(Sα)] for all α.
[Hint: Show that each of these conditions implies that h(Sα) is a section of E; conclude that it must be the section by h(α).]
(b) If E is a well-ordered set, show that no section of E has the order type of E, nor do two different sections of E have the same order type. [Hint: Given J, there is at most one order-preserving map of J into E whose image is E or a section of E.]
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