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Consequences of the Axiom of Choice
Paul Howard, Eastern Michigan University, Ypsilanti, MI, and Jean E. Rubin, Purdue University, West Lafayette, IN
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Mathematical Surveys and Monographs
1998; 432 pp; hardcover
Volume: 59
ISBN-10: 0-8218-0977-6
ISBN-13: 978-0-8218-0977-8
List Price: US$108 Member Price: US$86.40
Order Code: SURV/59

This book, Consequences of the Axiom of Choice, is a comprehensive listing of statements that have been proved in the last 100 years using the axiom of choice. Each consequence, also referred to as a form of the axiom of choice, is assigned a number.

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.

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

"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