Consequences of the Axiom of Choice (Mathematical Surveys and Monographs) (英語) ハードカバー – 1998/7
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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. It features a complete summary of all the work done in the last 100 years on statements that are weaker than the axiom of choice software provided. It 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, and 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. 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.
Another amazing thing is the large number of choice axioms. I had known about four or so choice axioms. In the section on "choice forms", there are roughly 80 more-or-less independent choice axioms and choice-like selection principles. (Pages 91-103.) Including equivalent forms, there are about 158 choice axioms and choice-like principles.
Then there are the 52 Cohen models for ZF set theory and the 64 Fraenkel-Mostowski permutation models for ZF with atoms. For each model, the known truth or falsity of 383 "forms" are indicated, together with extensive references and much useful explanation in the 148 Notes (pages 229-318). For example, the known equivalences between definitions of finite sets are stated, and the known truth or falsity of choice axioms and numerous choice-dependent theorems in algebra, graph theory, topology, logic and cardinal numbers.
Much of the book gives the impression of being a database rather than a normal book, which is obviously not unintentional. A typical book about model theory presents half a dozen models. So the 116 models in this book should be enough for anyone. (It doesn't present simple models like the constructible universe and von Neumann universe, however.)
The book is complemented by an online web site hosted by Eastern Michigan University. Using the book together with that web site, I was able to discover two Cohen models (the Solovay model and Shelah's Model II) for which the desired choice axioms were true and the undesired theorems were false.
This book brings together a vast amount of information about set theory models and their properties and relations which no 10 books on model theory could possibly give.