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Appendix on the Axiom of Choice

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Appendix on the Axiom of Choice Appendix A Appendix on the Axiom of Choice We repeat the following definition from Chapter 1. Definition A.1. A partial ordering on a nonempty set E is a relation ≤ on E such that 1) ∀ a ∈ E, a ≤ a,(reflexive property) and 2) ∀ a, b, c ∈ E, a ≤ b and b ≤ c ⇒ a ≤ c.(transitive property) We sometimes write b ≥ a for a ≤ b. We call a partial ordering antisymmetric if 3) ∀ a, b ∈ E, a ≤ b and b ≤ a ⇒ a = b. For an antisymmetric ordering we write a < b for a ≤ b but a = b. A partial ordering ≤ is called a total ordering if 4) ∀ a, b ∈ E, either a ≤ b or b ≤ a. A total ordering ≤ is called a linear ordering if it is antisymmetric. Recall that for a set E with a partial ordering ≤,amaximal element of E is an element z such that for each y ∈ E, y need not be related to z,butify ≥ z, then we also have y ≤ z. For an antisymmetric ordering, this means that y = z. { α ∈ I } As noted in Chapter 1, the Axiom of Choice states that if Sα : is a nonempty collection of nonempty sets, then there is a function T : I → α∈I Sα such that for every α ∈ I , T(α) ∈ Sα . If the sets are disjoint, we may think of a parliament. Recall the example given by Bertrand Russell in terms of pairs of shoes and pairs of socks: Given a finite or even countably infinite set of pairs of shoes, one can always pick one shoe from each pair, e.g., the left shoe. Given a finite collection of pairs of socks, one can pick one sock from each pair, but what happens with an infinite collection of pairs of socks? The Axiom of Choice says there is a set consisting of exactly one sock from each pair. We start with a result which, as we shall show, is equivalent to the Axiom of Choice. It is a modification of results and proofs in [17]. We will call a nonempty family C of subsets of a set S a chain if for any A and B ∈ C either A ⊆ B or B ⊆ A. We will let ∪C denote the union of the sets in a chain C . © Springer International Publishing Switzerland 2016 221 P.A. Loeb, Real Analysis, DOI 10.1007/978-3-319-30744-2 222 A Appendix on the Axiom of Choice Theorem A.1 (Hausdorff Maximal Principle for Linear Orderings). Let ≤ be a partial ordering on a nonempty set E. Let F0 be a nonempty subset of E that is linearly ordered with respect to ≤. There is a maximal (with respect to ⊆ ) subset F ⊆ E such that F0 ⊆ F and ≤ is a linear ordering on F. Proof. We let F be the family of all F ⊆ E such that the restriction of the ordering ≤ to F is a linear ordering and F ⊇ F0.LetT (∅)=F0, and for each F ∈ F ,let T(F) be an element of F that strictly contains F if there is one, and otherwise let T(F)=F. Here we have used the Axiom of Choice to define T. We want to show that there is an F ∈ F with T(F)=F. By the definition of T,thisF is a maximal linear ordered subset of E containing F0. Now, the union of the sets in a chain C consisting of members of F is again in F . To see this, let x and y be members of this union. We have x ∈ A ∈ C , and y ∈ B ∈ C , and either A ⊆ B or B ⊆ A. Suppose the first. Then both x and y are in B, so since ≤ is a linear ordering on B, either x ≤ y or y ≤ x; if both then x = y.The same is true if B ⊆ A. Let W be the collection of all chains C consisting of members of F such that for each F ∈ C with F F0, ∪{A ∈ C : A F}∈C , and for each F ∈ C such that F = ∪{A ∈ C : A F}, F = T (∪{A ∈ C : A F}). This means that T uses the Axiom of Choice to supply the next step in the containment after the union. Note that the singleton {F0} is in W . Given C ∈ W , we say that a chain D is an initial chain of C if D ∈ W , each F ∈ D is a member of the chain C , and if A ∈ C and A ⊂ F, then A ∈ D. The union G of a nonempty family of initial chains of C is an initial chain of C . To see this, note that F0 is in G , and each member of G is a member of C . Moreover, if A and B are members of G , then as elements of the chain C , either A ⊆ B or B ⊆ A.Also, each A ∈ G is an element of an initial chain D ∈ W ,soifB ∈ C and B ⊂ A, then B ∈ D.ItfollowsthatG ∈ W . If C and D are chains in W , then either C is an initial chain of D or D is an initial chain of C . To see this, let G be the union of all chains in W that are both initial chains of C and initial chains of D. The singleton {F0} is such a chain. As before, G ∈ W , and G is the largest initial chain of both C and D. Suppose C = G . Let F = ∪G .IfF ∈/ G ,letG = G ∪{F}.IfF ∈ G ,letG = G ∪{T (F)}. Then G ∈ W , and G is strictly larger than G initial chain of C . Similarly, if D = G , then G is strictly larger than G initial chain of D. Since we cannot have both, either G = C or G = D, or both. Now let G be the union of all chains in W . Then G ∈ W . To see this, note that the singleton {F0} is in G .IfA and B are in G , then A ∈ C ∈ W , and B ∈ D ∈ W . If C is an initial chain of D, then both A and B are in D, so either A ⊆ B or B ⊆ A. Similarly, if D is an initial chain of C , then both A and B are in C , so either A ⊆ B or B ⊆ A. Therefore, G is a chain. If F ∈ G , then F ∈ C for some C ∈ W . Therefore, if F = F0, ∪{A ∈ C : A F}∈C ⊆ G .IfF = ∪{A ∈ C : A F}, then F = T (∪{A ∈ C : A F}). Since G ∈ W and G is the union of all chains in W , ∪G ∈ G . Since G ∪{T (∪G )} must equal G , ∪G is a maximal linearly ordered subset of E containing F0. A Appendix on the Axiom of Choice 223 Remark A.1. A similar proof, left to the reader, establishes the Hausdorff Maximal Principle for total orderings. Theorem A.2. The following principles are equivalent: 1) Axiom of Choice. 2) Hausdorff Maximal Principle for Linear Orderings. 3) Zorn’s Lemma: Let ≤ be a partial ordering on a nonempty set E.Ifevery linearly ordered subset of E has an upper bound in E, then there is a maximal element with respect to ≤ contained in E. 4) Well-Ordering Principle: Every set can be ordered with a well-ordering. Proof. 1) ⇒ 2): This is Theorem A.1. 2) ⇒ 3): By the assumption, we can find a maximum (under containment) lin- early ordered subset F of E.Letz be an upper bound of F. We will show that z is a maximal element of E. Consider any y such that z ≤ y. We must show that we also have y ≤ z; they do not have to be equal. Now since z is an upper bound of F,for each x ∈ F, x ≤ z ≤ y.SoF ∪{y} is totally ordered with respect to ≤. Since F is a maximal linearly ordered set in E, it must also be true that for some x ∈ F, y ≤ x, whence, y ≤ z. 3) ⇒ 4): Given a nonempty set E, consider subsets F of E with a well-orderings ≤F .LetF be the set of all such pairs (F,≤F ). (A given set may appear more than once but with different orderings. In this sense, the subscript on the ordering is a bit misleading.) We can put an antisymmetric, partial ordering on F by setting (F,≤F ) (G,≤G) when i) F ⊆ G, ii) ≤G=≤F on F × F, [i.e., if x, z ∈ F, then x ≤F z, if and only if x ≤G z], and iii) for every pair (x,y) with x ∈ F and y ∈ G \ F, x ≤G y. We wish to show that is, in fact, an antisymmetric, partial ordering on F . Clearly, for each element (F,≤F ) in F , (F,≤F ) (F,≤F ) [(iii) is vacuously satis- fied]. Moreover, (F,≤F ) (G,≤G) and (G,≤G) (H,≤H ) ⇒ (F,≤F ) (H,≤H ). To see this, we note that Properties i and ii are clear, so we only check (iii). If y ∈ H \ F and x ∈ F, then for the case y ∈ H \ G,wehavex ≤H y since x ∈ G.If y ∈ G\F, then we have x ≤G y,sox ≤H y by (ii). To see that is antisymmetric, we assume that we have both (F,≤F ) (G,≤G) and (G,≤G) (F,≤F ).By(i),F = G. By (ii), ≤G= ≤F . Fix a subset G = {(Fα ,≤F )} of F , linearly ordered with respect to .Weset G = ∪α Fα , and we define an ordering on G by setting x ≤G y if x ≤Fα y for some α.
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