**Mathematical Surveys and Monographs**

Volume: 59;
1998;
432 pp;
Hardcover

MSC: Primary 03;

Print ISBN: 978-0-8218-0977-8

Product Code: SURV/59

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

**Electronic ISBN: 978-1-4704-1286-9
Product Code: SURV/59.E**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

#### Supplemental Materials

# Consequences of the Axiom of Choice

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*Paul Howard; Jean E. Rubin*

This book,

Part I is a listing of the forms by number. In this
part each form is given together with a listing of all statements
known to be equivalent to it (equivalent in set theory without the
axiom of choice). In Part II the forms are arranged by topic. In
Part III we describe the models of set theory which are used to show
non-implications between forms. Part IV, the notes section, contains
definitions, summaries of important sub-areas and proofs that are not
readily available elsewhere. Part V gives references for the
relationships between forms and Part VI is the bibliography.

Part VII is contained on the floppy disk which is enclosed in the
book. It contains a table with form numbers as row and column
headings. The entry in the table in row \(n\), column \(k\) gives the status
of the implication “form \(n\) implies form \(k\)”. Software for
easily extracting information from the table is also provided.

Features:

- complete summary of all the work done in the last 100 years on statements that are weaker than the axiom of choice
- software provided gives complete, convenient access to information about relationships between the various consequences of the axiom of choice and about the models of set theory
- descriptions of more than 100 models used in the study of the axiom of choice
- an extensive bibliography

About the software: Tables 1 and 2 are accessible on the PC-compatible software included with the book. In addition, the program maketex.c in the software package will create &tex; files containing copies of Table 1 and Table 2 which may then be printed. (Tables 1 and 2 are also available at the authors' Web sites: http://www.math.purdue.edu/\(\sim\)jer/ or http://www.emunix.emich.edu/\(\sim\)phoward/.) Detailed instructions for setting up and using the software are included in the book's Introduction, and technical support is available directly from the authors.

#### Readership

Mathematicians with a primary interest in set theory and who do research on the axiom of choice; philosophers interested in the role of the axiom of choice in the foundations of mathematics.

#### Reviews & Endorsements

This volume is a veritable encyclopedia, produced with impeccable scholarship, surveying the vast literature of theorems whose proof requires some form of a choice principle. This is the subject matter of this monograph in which the reader can find with ease the required information, including all pertinent definitions and references to the literature. The book also contains some new results, with proofs supplied in the section of notes. All in all, the authors and the publisher are to be congratulated for having produced such an outstanding monograph.

-- Mathematical Reviews