Review: Bruno Poizat, Cours de Theorie des Modeles. Une Introduction a la Logique Mathematique Contemporaine

Possibly the most important section of the entire book is the first section of Chapter XIII. Here the author gives a partial answer to what kind of structure theory is available for superstable theories without DOP and with PMOP. Call such a theory classifiable. The result in this section is that if X > \T\ + 2 and M and N are two models of a classifiable theory of cardinality X that are J^-equivalent, then M = N. K complementary result is available and it is a pity that it was not published here. It states that if T is countable and not classifiable, then for every regular I > 2"° there are models of cardinality X that are ifMA-equivalent but not isomorphic (see Shelah, Existence of many Laj ^-equivalent, nonisomorphic models of T of power X, Annals of pure and applied logic, vol. 34 (1987), pp. 291-310). These results reflect greatly on the absoluteness of non-isomorphism but to see exactly how, it is best to translate these results into the language of forcing. Fix a regular cardinal X greater than 2. Call a forcing A-structural if it preserves cardinals less than or equal to X and adds no new subsets of X of smaller cardinality. For instance, A-closed forcings are 2-structural. Any back-and-forth system of partial isomorphisms witnessing . 2^-equivalence is Xstructural and so the main result of Section 1 can be translated: Non-isomorphism between models of cardinality X of a countable classifiable theory is absolute for A-structural forcings. The results of Shelah's 1987 paper, on the other hand, say that this non-isomorphism relation in the case of a nonclassifiable theory is not absolute for A-structural forcings. It is possible that the author has reached a thesis for the meaning of structure by the end of the 1987 paper: T has a structure theory if and only if non-isomorphism between models of T of cardinality X is absolute for all A-structural forcings. Certainly it seems reasonable that if non-isomorphism is not suitably absolute then no structure theory is possible. If there was any doubt about whom the new edition is intended for, one need look no further than the first section of the book, the problem section. Most of the problems cannot be understood without having read sizable portions of the book. Be forewarned that some of the problems have been solved and one should consult at least the Added in proof section before working on them. The comments found there pertain mostly to solutions of the problems since 1983. Items 18 through 23 of this section may appear a bit strange until one realizes that this is an introduction to another work by the author entitled Universal classes (LV 878). Many of the problems that remain open are quite interesting. Just to mention one, the analogue of the Main Gap for uncountable theories is open. The author asks for a resolution of this analogue, although he probably expects the same sort of structure/non-structure dichotomy as discussed above. Readers of the problems section will enjoy the author's comments on Vaught's conjecture found at the end of page xxi. Conclusion. This book is very hard to read but the four new chapters are easier to read than the first nine. It is unfortunate that the last four chapters were not published separately as a research monograph. This might have led to earlier publication and a less expensive book. Overall though, this book is an absolute necessity for anyone currently studying classification theory. It contains the only published proof that unsuperstable theories have many models, most of the known results about the number of pairwise non-embeddable models of a first-order theory, and the structural half of the "5^,,-equivalent implies isomorphic" dichotomy. Any serious student of classification theory must come by the knowledge of the material in this book somehow, and although the book can be trying (both for what it contains and what it does not), its study pays off in the long run. The problem section contains many problems of dissertation quality and it guarantees that the book will not be the last word written on classification theory. BRADD HART