The Axiom of Choice is the statement that, for each family of nonempty sets, there is a function \(f\) from the family to its union such that \(f(S)\in S\) for each set \(S\) in the family. Thomas Jech’s The Axiom of Choice is, in its Dover edition, a reprint of the 1973 classic which explains the place of the Axiom of Choice in contemporary mathematics, that is, the mathematics of 1971–1972. The book contains problems at the end of each chapter of widely varying degrees of difficulty, often providing additional significant results. Each chapter also includes a paragraph or two of historical remarks, telling only who introduced certain ideas and who provided the proofs.
In Chapter 1, Jech provides a few examples of “unpleasant” consequences of the Axiom of Choice. The first is Vitali’s construction of a set that is non-Lebesgue measurable. (I use the term “construction” even though some mathematicians would argue that this is not a construction because of the use of arbitrary choices.) Next is Hausdorff’s construction of a decomposition of a sphere into a countable subset and three congruent subsets \(A\), \(B\), and \(C\) such that, in addition, \(A\cup B\) is also congruent to \(A\), \(B\), and \(C\). The chapter concludes with the result of Banach and Tarski that a solid ball can be decomposed into a finite number of parts that can be reassembled (using only rotations and reflections) into two balls, each congruent to the original one.
As a counterpoint, Chapter 2 provides a few examples of equivalences and consequences of the Axiom of Choice that are fundamental in other areas of mathematics. Jech proves that equivalences to the Axiom of Choice include Zermelo’s Well-Ordering Principle (all sets can be well-ordered), Zorn’s Lemma (if, in a non-empty partially ordered set, every chain has an upper bound, then the set has a maximal element), and Tukey’s Lemma (a nonempty family of sets which has finite character has a maximal element with respect to inclusion). Among the other consequences are Tychonoff’s Theorem (the product of compact spaces is compact in the product topology); the Hahn-Banach Theorem; that every vector space has a basis; and that every field has a unique algebraic closure up to isomorphism. Other consequences, for which Jech also includes a discussion of their strength relative to the Axiom of Choice, are the Prime Ideal Theorem (for Boolean algebras), the Consistency Principle, the Compactness Theorem for first-order logic, the Ultrafilter Theorem, the Stone Representation Theorem, and the Artin-Schreier Theorem.
The next group of results discussed are related to the Countable Axiom of Choice; here the results include that the set of real numbers is not a countable union of countable sets, the equivalence of sequentially defined topological properties of the real line with the epsilon-delta defined properties, and the countable additivity of Lebesgue measure. The chapter ends with a discussion of cardinal number arithmetic and Dedekind’s definition of finite sets.
The third chapter shows the consistency of the Axiom of Choice with the other Zermelo-Fraenkel axioms for set theory (ZF), using Gödel’s model of the constructible universe. In order to develop the model theory that Jech will use for showing the independence of the Axiom of Choice (by showing the consistency of the negation of the Axiom of Choice with ZF), he introduces permutation models for set theory with atoms. (Atoms are objects in a model which are not themselves sets: atoms have no elements.) Models constructed here include the first (respectively, second) Fraenkel model; this provides a proof that the Axiom of Choice (respectively, the Axiom of Choice for countable families of set doubletons) is independent from the axioms of ZF for set theory with atoms. The ordered Mostowski model shows that for set theory with atoms, the Axiom of Choice is independent from the Ordering Principle (every set can be linearly ordered).
In Chapter 5, Jech discusses generic models, Cohen’s method of forcing, and symmetric submodels of generic models. Various models of ZF show the following results: the Axiom of Choice is unprovable in ZF, so that the Axiom of Choice is independent from ZF; the Axiom of Choice for countable families of set doubletons is unprovable in ZF; and the Axiom of Choice is independent from the Ordering Principle.
Chapter 6 points out the analogies between permutation models of ZF with atoms and symmetric models of ZF. In some circumstances (but not all), it is possible to use this chapter’s first and second embedding theorems to transfer results from permutation models to symmetric models. These ideas are used in Chapter 7 to assist with the proofs that the Axiom of Choice is independent from the Prime Ideal Theorem, that the Prime Ideal Theorem is independent from the Ordering Principle, and that the Ordering Principle is independent from the Axiom of Choice for families of finite sets. The final section considers the statements C-n, namely the existence of choice functions for families of sets with at most a finite number n of elements, and the Axiom of Choice for families of finite sets. Jech shows that even if C-n holds for every finite n, the Axiom of Choice for families of finite sets can be unprovable.
The weaker versions of the Axiom of Choice discussed in Chapter 8 include aleph-restricted variations of the Principle of Dependent Choices and of the Axiom of Choice, and the comparability of the alephs with the cardinality of arbitrary sets.
Chapter 9 discusses some results that do not transfer from models of ZF with atoms to ZF. There are four results in particular that follow from the Axiom of Choice in ZF set theory with atoms and are each equivalent to the Axiom of Choice in ZF set theory, but such that the Axiom of Choice is independent from each of them in ZF set theory with atoms. These are:
1.The Axiom of Multiple Choice: for each family of nonempty sets, there is a function f such that is a nonempty finite subset of S for each set S in the family;
2.The Antichain Principle: Each partially ordered set has a maximal subset of mutually incomparable elements;
3.Every linearly ordered set can be well-ordered; and
4.The power set of every well-ordered set can be well-ordered.
Chapters 10 and 11 look at some areas of mathematics if the Axiom of Choice fails. Here are only a few of the results. For the real line, one result is that there is a subset \(S\) of real numbers and a real number that is in the closure of \(S\) but for which there is no sequence of elements of \(S\) that converge to it. There is a function from the reals to the reals that is not continuous at a point but is sequentially continuous there. There is a set of real numbers that is neither closed nor bounded but every sequence of points in the set has a convergent subsequence. It is possible for a vector space to fail to have a basis. A subgroup of a free group may not itself be free. A field may fail to have an algebraic closure. Not only may the cardinality of sets fail to be linearly ordered, but in fact any partially ordered set can be represented by the ordering of the cardinality of some family of sets. After a discussion of results in the theory of the arithmetic of cardinal numbers, Chapter 11 concludes with a proof that the Generalized Continuum Hypothesis implies the Axiom of Choice.
The book concludes in Chapter 12 with properties that contradict the Axiom of Choice. One possibility is related to the existence of various kinds of large cardinal numbers. For example, in ZF, one can consistently assume that aleph-one is a measurable cardinal. An alternative to the Axiom of Choice is the Axiom of Determinateness, assuming that one or another of the two players of a particular kind of game in which players alternately select natural numbers has a winning strategy for one of the two players (see page 176 for details). Assuming the Axiom of Determinateness is sufficient to prove that every set of real numbers is Lebesgue measurable and that aleph-one is a measurable cardinal.
The Dover edition is nicely done and quite affordable. One would have to look elsewhere, of course, for the latest results in this field, as there have been significant advances in the past 40 years.
Joel Haack is Professor of Mathematics at the University of Northern Iowa.