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Fenstad Jens E.. General recursion theory. An axiomatic approach. Perspectives in mathematical logic. Springer-Verlag, Berlin, Heidelberg, and New York, 1980, XI + 225 pp.

setting. Most impressive are the uniqueness theorem IV.5.6' (for a recent simplification, see Shelah's paper in this JOURNAL, vol. 44 (1979), pp. 215-220) and the characterization theorems IV.4.14 and IV.4.18. Chapter V deals with stable theories only. Regular types are introduced and studied, weights assigned to all types in superstable theories and to the so-called semiregular ones in general. Uniand multidimensionality of theories are defined and a study of the number and structure of sufficiently saturated models begun. Two-cardinal theorems are proved. Chapter VI studies ultraproducts and emphasizes the connections between this book's classification of theories and the one by Keisler which uses ultraproducts. Chapter VII greatly refines the Ehrenfeucht-Mostowski method of constructing models generated by indiscernibles; full use of this is made in Chapter VIII. Chapters VIII and IX contain the solutions of many of the outstanding conjectures that motivated the whole theory. The important result, already mentioned, about the number of models of a superstable theory is just a particular case of stronger results concerning pseudo-elementary classes (see sections 2-3, Chapter VIII; the proofs are very intricate and contain many ingenious steps). Theorem VIII.4.7 establishes under what conditions a theory T has a saturated model of power X. Theorem IX. 1.15 implies Eos's conjecture 2, and IX.2.2 is the Baldwin-Lachlan result settling Vaught's conjecture 3. Other important conjectures are settled in IX.1.18, IX.1.19 and IX.2.1 (IX.2.1(1) is a result of Lachlan). Finally, the structure of sufficiently saturated models of superstable theories is further elucidated in IX.2.3 and IX.2.4.