''Equivalents of the Axiom of Choice'' is a book in mathematics, collecting statements in mathematics that are true

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

holds. It was written by Herman Rubin and

Jean E. Rubin Jean Estelle Hirsh Rubin (October 29, 1926 – October 25, 2002) was an American mathematician known for her research on the axiom of choice. She worked for many years as a professor of mathematics at Purdue University. Rubin wrote five books: thre ...

, and published in 1963 by North-Holland as volume 34 of their Studies in Logic and the Foundations of Mathematics series. An updated edition, ''Equivalents of the Axiom of Choice, II'', was published as volume 116 of the same series in 1985.

Topics

At the time of the book's original publication, it was unknown whether the

followed from the other axioms of

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

(ZF), or was independent of them, although it was known to be

consistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...

with them from the work of

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

. This book codified the project of classifying theorems of mathematics according to whether the axiom of choice was necessary in their proofs, or whether they could be proven without it. At approximately the same time as the book's publication,

Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a F ...

proved that the negation of the axiom of choice is also consistent, implying that the axiom of choice, and all of its equivalent statements in this book, are indeed independent of ZF. The first edition of the book includes over 150 statements in mathematics that are equivalent to the axiom of choice, including some that are novel to the book. This edition is divided into two parts, the first involving notions expressed using sets and the second involving classes instead of sets. Within the first part, the topics are grouped into statements related to the

well-ordering principle In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set of nonnegative integers is well-ordered by its "natural" or "magnitude" order in which x pr ...

, the axiom of choice itself,

trichotomy A trichotomy can refer to: * Law of trichotomy, a mathematical law that every real number is either positive, negative, or zero ** Trichotomy theorem, in finite group theory * Trichotomy (jazz trio), Australian jazz band, collaborators with Dan ...

(the ability to compare

cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...

s), and

Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...

and related maximality principles. This section also includes three more chapters, on statements in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, statements for

s, and a final collection of miscellaneous statements. The second section has four chapters, on topics parallel to four of the first section's chapters. The book includes the history of each statement, and many proofs of their equivalence. Rather than ZF, it uses

Von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collec ...

for its proofs, mainly in a form called NBG⁰ that allows

urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set (has no elements), but that may be an element of a set. It is also referred to as an atom or individ ...

s (contrary to the

axiom of extensionality The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the ...

) and also does not include the

axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoin ...

. The second edition adds many additional equivalent statements, more than twice as many as the first edition, with an additional list of over 80 statements that are related to the axiom of choice but not known to be equivalent to it. It includes two added sections, one on equivalent statements that need the axioms of extensionality and regularity in their proofs of equivalence, and another on statements in

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, and

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

. It also includes more recent developments on the independence of the axiom of choice, and an improved account of the history of Zorn's lemma.

Audience and reception

This book is written as a reference for professional mathematicians, especially those working in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

. Reviewer

Chen Chung Chang Chen Chung Chang () was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's co ...

writes that it "will be useful both to the specialist in the field and to the general working mathematician", and that its presentation of results is "clear and lucid". By the time of the second edition, reviewers J. M. Plotkin and David Pincus both called this "the standard reference" in this area.

References

{{reflist, refs= {{citation, title=Review of ''Equivalents of the Axiom of Choice'', last=Chang, first=C.-C., authorlink=Chen Chung Chang, journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr=0153590 {{citation, title=Review of ''Equivalents of the Axiom of Choice'', first=R. L., last=Goodstein, authorlink=Reuben Goodstein, journal=

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, volume=48, issue=365, date=October 1964, page=348, doi=10.2307/3613069, jstor=3613069 {{citation, first=J. M., last=Plotkin, title=Review of ''Equivalents of the Axiom of Choice, II'', journal=

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ...

, zbl=0582.03033 {{citation, title=Review of ''Equivalents of the Axiom of Choice, II'', first=David, last=Pincus, journal=

Journal of Symbolic Logic The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zent ...

, volume=52, issue=3, date=September 1987, pages=867–869, doi=10.2307/2274372, jstor=2274372 {{citation, title=Review of ''Equivalents of the Axiom of Choice, II'', last=Smith, first=Perry, year=1987, journal=

, mr=0798475

External links

''Equivalents of the Axiom of Choice, II''
at the

Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

Axiom of choice Mathematics books 1963 non-fiction books 1985 non-fiction books