Consequences of the Axiom of Choice, Band 1
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.
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Abelian group AC AC antichain Assume axiom of choice Baire Blass Boolean algebra Brunner C(oo C(WO cardinal choice function clear compact countable Dedekind finite set Dedekind finite subset define denumerable family denumerable set Dependent Choices disjoint union element equivalent finite sets FM model follows form 14 form 43 form known forms at least forms implying function f Harper/Rubin 1976 Hickman homomorphism Howard Howard/Yorke 1989 iff AC implies AC implies form implying each form infinite set Jech Keremedis 1996d lattice lemma Lindelof linear list of forms Math metric space minimal list Moore Morillon Mostowski non-empty sets note 18 N3T note 71 note 94 order automorphism ordered subset ordinal pairwise disjoint partially ordered set partition permutation model Pincus prime ideal prob proof Rubin Rubin/J satisfies sequence set of atoms subgroup T2 space topological space true Truss ultrafilter upper bound vector space
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