Abstract

This is a very high quality book with a slightly misleading title. It is difficult to see how it could serve as an introduction for anyone except the mathematically mature or, at least, a student who has already been introduced to formal logic through the lower predicate calculus. Not that these topics are not covered in the book—they comprise the first 92 pages; but the discussion quickly moves into intellectual high gear with sophisticated treatments of the independence of systems of propositional logic, first-order theories, isomorphism of interpretations, and completeness and decidability. With this caveat, Mendelson's book is one of the best of the "second generation" of books on mathematical logic. It is lucid and rigorous, and contains no "frills." It does, however, contain numerous exercises in which the student will have the pleasure of developing for himself some of the more important theorems in the various subjects treated. Among these are chapters on formal number theory, including Gödel's incompleteness theorem, recursive undecidability, Tarski's theorem; a chapter on axiomatic set theory with Hartog's theorem, the axiom of choice, and the axiom of restriction ; and a truly excellent chapter on effective computability which takes off from the notion of an algorithm, and then uses this as a basis upon which to develop the theories of Markov algorithms, Turing algorithms, the theories of Herbrand and Gödel, recursively enumerable sets and undecidable problems. An appendix contains a version of K. Schütte's consistency proof for formal number theory.—H. P. K.