Mathematical logic. A course with exercises. Part II. Recursion theory, Gödel's theorems, set theory, model theory. Translated from the 1993 French original by Donald H. Pelletier (Q2734834)

This is the second part of a two-volume course in mathematical logic. The first part (2000; Zbl 0955.03004) [for reviews of the original French edition see Zbl 0787.03001 and Zbl 0787.03002] dealt with the propositional calculus, Boolean algebra, predicate calculus, and the completeness theorems. In this second part, the authors maintain their high standard of precision, along with adequate motivation for sufficiently sophisticated readers. The core of the subject is covered very well. The chapter on recursion theory contains the central results on recursive functions, Turing machines, and recursively enumerable sets. The chapter on incompleteness and undecidability is standard, but very neat and clear. The chapter on set theory, based on the Zermelo-Fraenkel system, gets as far as inaccessible cardinals and the reflection scheme, and includes a few well-chosen relative consistency theorems. The chapter on model theory contains the interpolation and definability theorems, a clear treatment of reduced products and ultraproducts, a few ``preservation'' theorems, and a section on aleph-nought categorical theories. Especially noteworthy is a 95-page Appendix of solutions of every exercise in the book.