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The Axiom of Choice and Some Equivalences: a Senior Exercise

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The Axiom of Choice and Some Equivalences: A Senior Exercise Tate Deskins Department of Mathematics Kenyon College November 29, 2012 1 CONTENTS Contents 1 Introduction and Motivation 3 1.1 Overview . .3 1.2 An Introduction to Set Theory . .3 1.3 Axiom of Choice . .4 2 Axiom of Choice and Well-Ordering 5 2.1 Choice Functions, Orderings et al. .5 2.2 Ordinal Numbers . .6 2.3 Induction. .9 2.4 Recursion . 10 2.5 Hartog's Number . 14 2.6 Well-Ordering Principle . 14 2.7 Implications? . 15 3 Maximality 16 3.1 Maximally Basic . 17 3.2 Well-Ordering to Zorn's Lemma . 19 3.3 Zorn's Lemma to Tukey's Lemma . 19 3.4 Implications? . 21 4 Equivalence 22 4.1 Functions . 22 4.2 Back to Choice . 22 5 Concluding Remarks 23 Choice References 25 2 1 INTRODUCTION AND MOTIVATION 1 Introduction and Motivation 1.1 Overview Frequently in our undergraduate math courses a professor, while lecturing or helping a student through a proof, will give a sentence to this effect: \This step implicitly uses the Axiom of Choice." Some students nod knowingly, while others wonder, \What is the Axiom of Choice?" For those of us who fall into the latter category, hopefully this paper will answer that question and demonstrate some of the powerful consequences of the Axiom of Choice. The plan is to first motivate why we should care about the Axiom of Choice and explain its statement. Following this we will show in a round- about, yet elegant way, the equivalence of the Axiom of Choice to the Well- Ordering Principle, Zorn's Lemma, and Tukey's Lemma. After each step of the proof we will discuss some of the consequences of the statements. 1.2 An Introduction to Set Theory Axiomatic set theory is an attempt by mathematicians to describe mathe- matics starting with a primitive, the set, and a list of rules, the axioms, which govern how we are allowed to manipulate the primitive. We shall rely on our intuition from Foundations for what a set is. An axiom as stated dictates how we can manipulate sets. Here are a couple of examples of axioms for ZF1 set theory2: Axiom 1.1 (Extensionality): If two sets have exactly the same members then they are equal. Formally stated: 8X8Y (8a(a 2 X , a 2 Y ) ) X = Y ): Axiom 1.2 (Empty Set): There exists a set with no members: 9X8a(a 62 X): Axiom 1.3 (Pairing): For any sets X and Y there is a set containing only X and Y : 8X8Y 9Z8a(a 2 Z , a = X or a = Y ): 1ZF stands for Zermelo-Fraenkel, the last names of the two mathematicians who are attributed with creating the axioms of this set theory. 2Some of the statements of these axioms are taken verbatim and others paraphrased from [1]. 3 1 INTRODUCTION AND MOTIVATION There are several more axioms in standard ZF set theory that allow us to take unions, make power sets, and construct subsets [1]. However, in order to avoid having to retitle this document \Basic Axiomatic Set Theory," we shall assume that you have a knowledge of the basic set theory that one would learn from Foundations. 1.3 Axiom of Choice Given the basic axioms we can describe much of the mathematics that we know in set theoretic terms; however, there is another axiom that is needed to build some of the major theorems (or even what we may think of as obvious facts) of mathematics. This axiom is the Axiom of Choice. In essence, the Axiom of Choice allows us to select members from collections of nonempty sets, i.e. pull representatives from a partition, or create functions from arbi- trary relations with the same domain. These seem like very reasonable, and necessary, things to be able to do. Since axioms are something that we base on our intuition, we are tempted to accept the Axiom of Choice as an axiom. In general, Mathematicians find the Axiom of Choice too useful to ignore and thus include it as one of the Axioms of set theory. Let us now give the statements of the Axiom of Choice and some of its equivalents: Axiom of Choice 1 (Axiom of Choice): Every set has a choice function [1, 3, 4, 5, 6]. Axiom of Choice 2 (Well-Ordering Principle): All sets can be well-ordered [1, 3, 4, 5, 6]. Axiom of Choice 3 (Zorn's Lemma): If X is a partially ordered set where each chain has an upper bound, then X has a maximal element [2, 3, 5]. Axiom of Choice 4 (Tukey's Lemma): If a collection of sets U is of finite character then U contains maximal sets under the ordering \⊆" [2, 3, 5]. Before we dive in there is some terminology to discuss. We shall develop and define the mathematics necessary as the need arises. 4 2 AXIOM OF CHOICE AND WELL-ORDERING AC1 AC4 AC2 AC3 Figure 1: The path for our proof of some equivalent statements of the Axiom of Choice. 2 Axiom of Choice and Well-Ordering Before we give the proof of the Well-Ordering Principle we have several math- ematical concepts to develop. 2.1 Choice Functions, Orderings et al. Colloquially, a choice function will pick an element out of a set. However since a choice function can not pick an element out of an empty set (there is nothing in there) it is useful at times to define for a set S, F(S) = P(S)nf;g. Where P(S) is the power set of S. Definition 2.1: Let S be a set of nonempty sets. A function f is a choice function on S if [ f : S ! T; T 2S such that f(T ) 2 T for all T 2 S [5]. Notationally we shall use Dom(f) and Ran(f) to be the domain and range of a function f. 5 2 AXIOM OF CHOICE AND WELL-ORDERING There are three types orderings that we may recall from Foundations3: Definition 2.2 (Partial Ordering): Given a set S, a relation \ v" is a partial ordering if: • for s; t 2 S s v t and t v s then s = t • for s; t; r 2 S, if s v t and t v r then s v r • for all s 2 S s v s. The set and ordering in Figure 2 is a partial ordering. Definition 2.3 (Total Ordering): Given a set S, a relation \E" is a total ordering if: • (S; E) is a partial ordering • for all s; t 2 S, s E t or t E s. Definition 2.4 (Well-Ordering): Given a set S, a relation \" is a well- ordering if: • (S; ) is a total ordering • for all T ⊆ S, there is a t 2 T such that t s for all s 2 T . With choice functions and some orderings under our belt, let us talk about ordinals. 2.2 Ordinal Numbers Let us define the successor of a set X as S(X) = X [ fXg [4]. With this tool we will consider a different picture of the natural numbers: 0 = ; 1 = S(0) = f0g 2 = S(1) = f0; 1g 3 = S(2) = f0; 1; 2g . 3The following three definitions were paraphrased directly from [7]. 6 2 AXIOM OF CHOICE AND WELL-ORDERING p1 p4 m u p2 p3 Figure 2: An example of a partially ordered set (by \rightness" on the page). The dashed nodes are part of a chain of which m is a maximal element and u an upper bound. Thus for a non-zero natural number n 2 N we define it recursively4 as n = S(n − 1). We can extend this definition of the natural numbers further by defining ! as the union of all natural numbers (finite ordinals), i.e. ! = N. We can then continue defining what we will call the ordinal numbers similarly: ! = N ! + 1 = S(!) = N [ f!g ! + 2 = S(! + 1) = N [ f!; ! + 1g ! + 3 = S(! + 2) = N [ f!; ! + 1;! + 2g . As we continue creating ordinals we will get 2! and ! · ! then eventually !!. Now just as we stepped beyond the natural numbers by taking the union of all natural numbers and defining this as !, we will define !1 to be the union of all the ordinals described by combinations of ! and the natural numbers. We can similarly define: !2;!3;:::;!!;!!+1;:::;!!1 ;::: So we have two different kinds of ordinal numbers: limit ordinals, like the list above, which are created by taking unions of all previous types of ordinals; and successor ordinals, such as ! + 9, which are created using the successor function. 4We will discuss recursion in more detail in Section 2.4 7 2 AXIOM OF CHOICE AND WELL-ORDERING It appears that the ordinal numbers are well-ordered under inclusion \⊆." This is in fact a theorem, but its proof is beyond the scope of the paper. Let us now formalize this intuitive understanding of the ordinal numbers. Definition 2.5: A set α is an ordinal number if: • every element of α is a subset of α • α is strictly well-ordered by \2" [4]. Definition 2.6: The ordinal α is a successor ordinal if there is an ordinal β such that S(β) = α. Otherwise α is a limit ordinal [4]. Lemma 2.7: Set inclusion \⊆" is a well-ordering on ordinal numbers. A proof of this lemma can be seen on page 109 of [4]. Another important fact about the ordinals is that they are not a set.
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