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 X be a set; let  be the collection of all pairs (A, <), where A is a subset of X and < is a well-ordering of A. Define

if (A, <) equals a section of (A′, <′).

(a) Show that ≺ is a strict partial order on .

(b) Let  be a subcollection of  that is simply ordered by ≺. Define B′ to be the union of the sets B. for all (B, <) ∈ ; and define <′ to be the union of the relations <, for all (B, <) ∈ . Show that (B′, <′) is a well-ordered set.