##
**Axiom of choice.**
*(English)*
Zbl 1102.03049

Lecture Notes in Mathematics 1876. Berlin: Springer (ISBN 3-540-30989-6/pbk). xiv, 194 p. (2006).

In the preface the author writes: “This book is not written as a compendium, or a textbook, or a history of the subject…I hope however, that this monograph might find its way into seminars…An attempt has been made to keep the material treated as simple and elementary as possible. In particular no special knowledge of axiomatic set theory is required. However, a certain mathematical maturity and a basic aquaintance with general topology will turn out to be helpful.”

In the first chapter (“Origins”) some fundamental aspects of the Axiom of Choice (AC) are described, particularly its historical connections with the Well-Order Theorem of Zermelo and the Continuum Hypothesis (CH). Chapter 2 (“Choice principles”) deals with equivalents to the Axiom of Choice (Hausdorff’s Maximal Chain Condition, Zorn’s Lemma, Teichmüller-Tukey Lemma) and several concepts related to it, particularly the Axiom of Multiple Choice, Kurepa’s Maximal Antichain Condition, Ultrafilters etc. On page 18 the interdependences between the different principles are summarized in a diagram whose implications were proved before. Chapter 3 (“Elementary observations”) exhibits examples where the application of AC is hidden resp. unnecessary. Further, some concepts which are related to AC are dicussed. Chapter 4 (“Disasters without choice”) presents a lot of theorems (in cardinal arithmetic, order theory, algebra, and especially in topology) which are equivalent to AC, or to the Ultrafilter theorem, and investigates how these theorems behave when AC is replaced with other assumptions. Many of these which are provable in ZFC are lost in ZF; some can be saved with other assumptions. Chapter 5 (“Disasters with choice”) deals among others with the Cauchy equation and Hamel bases, and extensively with the paradoxical decomposition of spheres and related objects (Theorems of Hausdorff, Banach-Tarski, Robinson, Sierpiński). Most of the theorems are given with proofs, and each section is followed by a series of exercises. Several useful diagrams are included in the text. This is enriched with many quotations concerning opinions about AC and other things.

In the first chapter (“Origins”) some fundamental aspects of the Axiom of Choice (AC) are described, particularly its historical connections with the Well-Order Theorem of Zermelo and the Continuum Hypothesis (CH). Chapter 2 (“Choice principles”) deals with equivalents to the Axiom of Choice (Hausdorff’s Maximal Chain Condition, Zorn’s Lemma, Teichmüller-Tukey Lemma) and several concepts related to it, particularly the Axiom of Multiple Choice, Kurepa’s Maximal Antichain Condition, Ultrafilters etc. On page 18 the interdependences between the different principles are summarized in a diagram whose implications were proved before. Chapter 3 (“Elementary observations”) exhibits examples where the application of AC is hidden resp. unnecessary. Further, some concepts which are related to AC are dicussed. Chapter 4 (“Disasters without choice”) presents a lot of theorems (in cardinal arithmetic, order theory, algebra, and especially in topology) which are equivalent to AC, or to the Ultrafilter theorem, and investigates how these theorems behave when AC is replaced with other assumptions. Many of these which are provable in ZFC are lost in ZF; some can be saved with other assumptions. Chapter 5 (“Disasters with choice”) deals among others with the Cauchy equation and Hamel bases, and extensively with the paradoxical decomposition of spheres and related objects (Theorems of Hausdorff, Banach-Tarski, Robinson, Sierpiński). Most of the theorems are given with proofs, and each section is followed by a series of exercises. Several useful diagrams are included in the text. This is enriched with many quotations concerning opinions about AC and other things.

Reviewer: Egbert Harzheim (Köln)

### MSC:

03E25 | Axiom of choice and related propositions |

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

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