Equivalents of the axiom of choice. 2nd ed.

*(English)*Zbl 0582.03033
Studies in Logic and the Foundations of Mathematics, Vol. 116. Amsterdam - New York - Oxford: North-Holland. XXVIII, 322 p. $ 45.00; Dfl. 140.00 (1985).

In this second edition of their well known 1963 monograph [see Zbl 0129.006] the authors greatly add to their catalogue of equivalent forms of choice and they incorporate the extensive knowledge of ”choice like” principles that was the product of the post Paul Cohen set-theoretic renaissance. Most equivalences are proved in \(NBG^ 0\) (Von Neumann, Bernays, Gödel set theory with atoms but without the axiom of foundation). There is, however, a new section on statements whose equivalence requires extensionality and foundation. There is also a new section on forms from topology, analysis, and logic. The section on algebraic forms is expanded to include statements on vector spaces and groups. All sections have undergone growth. The earlier monograph of 157 pages is now a work of 330 pages.

Especially helpful to the casual reader are the lists of the various forms which appear in the text. These lists are at the end of the monograph. One such list, the list of forms related to the axiom of choice, gives some statements whose relative strengths are unknown. This offers opportunities for further research. No doubt there are equivalents of choice not mentioned by the authors, but their work is definitely the standard reference for would be discoverers of choice-equivalents to consult.

Contents. Part I - Set Forms: the well-ordering theorem, the axiom of choice, the law of trichotomy, maximal principles, forms equivalent to the axiom of choice under the axioms of extensionality and foundation, algebraic forms, cardinal number forms, forms from topology, analysis, and logic. Part II - Class Forms: the well-ordering theorem, the axiom of choice, maximal principles.

Especially helpful to the casual reader are the lists of the various forms which appear in the text. These lists are at the end of the monograph. One such list, the list of forms related to the axiom of choice, gives some statements whose relative strengths are unknown. This offers opportunities for further research. No doubt there are equivalents of choice not mentioned by the authors, but their work is definitely the standard reference for would be discoverers of choice-equivalents to consult.

Contents. Part I - Set Forms: the well-ordering theorem, the axiom of choice, the law of trichotomy, maximal principles, forms equivalent to the axiom of choice under the axioms of extensionality and foundation, algebraic forms, cardinal number forms, forms from topology, analysis, and logic. Part II - Class Forms: the well-ordering theorem, the axiom of choice, maximal principles.

Reviewer: J.M.Plotkin

##### MSC:

03E25 | Axiom of choice and related propositions |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |