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Building Models by Games

Wilfrid Hodges
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Russell Jay Hendel
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This is an excellent book for a 2nd or 3rd year graduate course in Model Theory, especially for those wishing to concentrate in this area. The book has numerous standard attributes of good books as well as several unusual features which make it stand out as excellent. We first list the three unusual features.

  • Unifying approach: The book introduces a general method for building infinite mathematical structures using a multi-step procedure. This multi-step procedure is formulated in the language of games. The procedure simplifies, motivates and unifies several well-known constructions.
  • Rich applications: To fully appreciate the book, numerous mathematical fields must be known; it is for this reason that I would target the book to 2nd or 3rd year graduate students. I list many of the applications in the book:
    1. (Exercise 3.1#2) “Show that each of the following classes is axiomatized by a universal-2 first-order theory and describe the class of models in each class: the classes of algebraically closed fields, commutative von Neumann regular rings, boolean algebras, integral domains, commutative rings with no non-zero nilpotent elements, distributive lattices with 0 and 1;
    2. The use of Hilbert’s Nullstellensatz to show that existentially closed fields coincide with the algebraically closed fields;
    3. (Boolean Algebras) The study of first order Horn theories; the construction of uncountable boolean algebras without uncountable chains and no uncountable pairwise incomparable elements.
    4. Heavy use of Group theory (e.g. Exercise 3.3#5: Show that in an existentially closed group every element is a commutator and the product of two elements of order 2.)
    5. Throughout the book, illustrations and exercises are formulated in terms of specific algebraic structures (e.g. Given a commutative ring A and an element a, is there an extension B containing A with an element b such that a2 b =a).
  • Friendly End-of-Chapter notes: For example at the end of Chapter 4 gives chapter and section references in old books for each of the exercises. As another example at the end of Chapter 1 (intro) the author sympathetically gives multiple references to brush up on model theory: “Change and Kessler is the authoritative text on first order model theory up to about 1970. But it is rather too encyclopedic for a quick introduction. For that I recommend the opening chapters of either Bell and Slomson or Sacks for those who know some logic; readers whose logical background is weaker might start with Bridge or Malitz.”(Many other references are given)

We now list the standard features one expects in a good book on Model Theory.

  • Exercises: Over 160 (quite allot for a 2nd year graduate text). The exercises range from routine to advanced and include a) filling small gaps in proofs b) applications c) known things d) major items of research. The author concedes “I can’t do all the exercises myself.”
  • Open problems: There are about 2 dozen open problems listed at the end of the book. Many of these could be subjects of doctoral dissertations.
  • Bibliography: There are about 400 references in the bibliography.
  • Forcing: The book covers 10 types of forcing including, Henkin, Henkin-Orey, Makkai, Lambda, Robinson (finite and infinite) 0, Q, Skolem, Keisler and Shelah forcing.
  • Digestible sections: There are 24 sections in 8 chapters. This book is therefore easily coverable in one (certainly 2) semesters.

Finally the mathematical exposition is clear and the author has a delightful (or queer, depending on your point of view) sense of humor. Chapter 1 has a whole section on use of “pictures”. In the references, the author attributes the use of various pictures in mathematical logic to about half a dozen authors. Another example of the author’s sense of humor is his citation from a building construction book on making bricks to introduce Chapter 2 which presents Games and Forcing.

Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, pedagogy theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.)


2. Games and Forcing
3. Existential Closure
4. Chaos or Regimentation
5. Classical Languages
6. Proper Extensions
7. Generalised Quantifiers
8. L(Q) in Higher Cardinalities
List of types of forcing
List of open questions