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**Building models by games.**
*(English)*
Zbl 0569.03015

London Mathematical Society Student Texts, 2. Cambridge etc.: Cambridge University Press. VI, 311 p. hbk: £25.00; $ 42.50; pbk: £7.95; $ 14.95 (1985).

”This book is... about a very general method for building models”. The method is an elaboration of Henkin’s completeness proof, incorporating such later additions as R. M. Smullyan’s ”abstract consistency properties” [J. Math. Soc. Japan 15, 129-133 (1963; Zbl 0118.249)] and P. J. Cohen’s idea of ”forcing” [Proc. Natl. Acad. Sci. USA 50, 1143-1148 (1963; Zbl 0192.044)]. The model is constructed in a canonical fashion from a set of sentences. The set of sentences is constructed in stages. This construction is cast as a 2-person game (with infinitely many moves) between a player \(\forall\) and a player \(\exists\), an adaptation of an idea of A. Ehrenfeucht [Fundam. Math. 49, 129-141 (1961; Zbl 0096.243)]. (Though ”the two players are really only a technical device for analyzing possible constructions”, the author makes them more concrete by creating one male and the other female, thus finding helpful employment for ”he” and ”she”.) Enforcible properties of the constructed model are those which player \(\exists\) can force the model to have by playing according to a ”winning” strategy. The main question, then, is what properties can be enforced?

Chapter 1, Preliminaries, reviews the necessary basic concepts from model theory. Chapter 2, Games and Forcing, explains the general construction. Chapter 3, Existential Closure, treats adjunction of elements, a generalization of the resultant, existentially closed groups (a strengthening of the notion of an algebraically closed group), and (finite) Robinson forcing. Chapter 4, Chaos and Regimentation, treats atomic models, finite-generic models, and existentially closed nilpotent groups of class 2. Chapter 5, Classical Languages, treats the omitting types theorem, saturation, and (infinite) Robinson forcing. While the chapters mentioned so far deal (mainly) with countable languages, the remaining chapters use languages of larger cardinality and set theoretic principles (such as GCH and \(\diamond_{\lambda})\) appear. Chapter 6, Proper Extensions, treats definable ultrapowers and a result of Baumgartner on uncountable Boolean algebras. Chapter 7, Generalized Quantifiers, and chapter 8, L(Q) in Higher Cardinalities, make use of languages with additional quantifiers: Mostowski (cardinality) quantifiers Q and the Magidor-Malitz (or Ramsey) quantifiers.

Each section has numerous exercises, extending the results and applying it to algebra (mainly group theory). Each chapter has a list of references to the origins of the results and (most of) the exercises. The book concludes with two lists, the first a List of [ten] Types of Forcing from the literature and how they fit into the book, the second a List of [twenty-four] Open Questions with comments.

The author claims that ”the book consists of a forty-minute talk which he gave”, and the mortal reader, glancing at the wealth of material the book contains, will conclude that it is not for him. Actually, the book is skilfully written with loving concern for the reader. Anyone interested in the border between model theory and algebra should own a copy.

Chapter 1, Preliminaries, reviews the necessary basic concepts from model theory. Chapter 2, Games and Forcing, explains the general construction. Chapter 3, Existential Closure, treats adjunction of elements, a generalization of the resultant, existentially closed groups (a strengthening of the notion of an algebraically closed group), and (finite) Robinson forcing. Chapter 4, Chaos and Regimentation, treats atomic models, finite-generic models, and existentially closed nilpotent groups of class 2. Chapter 5, Classical Languages, treats the omitting types theorem, saturation, and (infinite) Robinson forcing. While the chapters mentioned so far deal (mainly) with countable languages, the remaining chapters use languages of larger cardinality and set theoretic principles (such as GCH and \(\diamond_{\lambda})\) appear. Chapter 6, Proper Extensions, treats definable ultrapowers and a result of Baumgartner on uncountable Boolean algebras. Chapter 7, Generalized Quantifiers, and chapter 8, L(Q) in Higher Cardinalities, make use of languages with additional quantifiers: Mostowski (cardinality) quantifiers Q and the Magidor-Malitz (or Ramsey) quantifiers.

Each section has numerous exercises, extending the results and applying it to algebra (mainly group theory). Each chapter has a list of references to the origins of the results and (most of) the exercises. The book concludes with two lists, the first a List of [ten] Types of Forcing from the literature and how they fit into the book, the second a List of [twenty-four] Open Questions with comments.

The author claims that ”the book consists of a forty-minute talk which he gave”, and the mortal reader, glancing at the wealth of material the book contains, will conclude that it is not for him. Actually, the book is skilfully written with loving concern for the reader. Anyone interested in the border between model theory and algebra should own a copy.

Reviewer: G.Fuhrken

### MSC:

03C25 | Model-theoretic forcing |

03C60 | Model-theoretic algebra |

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

20F18 | Nilpotent groups |

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

03C80 | Logic with extra quantifiers and operators |

03C75 | Other infinitary logic |

91A05 | 2-person games |