A.1. THE HAMEL BASIS 2727

and ify ∈ S2 \S1 then x≤2 y for all x ∈ S1,

and if ≤1is the well order of S1 then the two orders are consistent on S1. Then observe that≺ is a partial order on F . By the Hausdorff maximal principle, let C be a maximal chainin F and let

X∞ ≡ ∪C .

Define an order, ≤, on X∞ as follows. If x, y are elements of X∞, pick S ∈ C such that x, yare both in S. Then if ≤S is the order on S, let x≤ y if and only if x≤S y. This definition iswell defined because of the definition of the order,≺. Now let U be any nonempty subset ofX∞. Then S∩U ̸= /0 for some S ∈ C . Because of the definition of ≤, if y ∈ S2 \S1, Si ∈ C ,then x ≤ y for all x ∈ S1. Thus, if y ∈ X∞ \ S then x ≤ y for all x ∈ S and so the smallestelement of S∩U exists and is the smallest element in U . Therefore X∞ is well-ordered.Now suppose there exists z ∈ X \X∞. Define the following order, ≤1, on X∞∪{z}.

x≤1 y if and only if x≤ y whenever x,y ∈ X∞

x≤1 z whenever x ∈ X∞.

Then letC̃ = {S ∈ C or X∞∪{z}}.

Then C̃ is a strictly larger chain than C contradicting maximality of C . Thus X \X∞ = /0and this shows X is well-ordered by ≤. This proves the lemma.

With these two lemmas the main result follows.

Theorem A.0.4 The following are equivalent.

The axiom of choice

The Hausdorff maximal principle

The well-ordering principle.

Proof: It only remains to prove that the well-ordering principle implies the axiom ofchoice. Let I be a nonempty set and let Xi be a nonempty set for each i ∈ I. Let X = ∪{Xi :i ∈ I} and well order X . Let f (i) be the smallest element of Xi. Then

f ∈∏i∈I

Xi.

A.1 The Hamel BasisA Hamel basis is nothing more than the correct generalization of the notion of a basis for afinite dimensional vector space to vector spaces which are possibly not of finite dimension.