Abstract

A prodigious amount of material is contained within the pages of this short book. The various chapters comprise a quick but rigorous survey of the main results presented in advanced level courses in mathematical logic. The accent here is on the development of proofs for theorems, and not upon topics in the philosophy of mathematics or in "foundational studies." This is not a weakness. No worthwhile investigation of the philosophy or foundations of mathematics can today take place except on the prior basis of such material as is so succinctly presented in this book. Lyndon has thus rendered a service to those who find that they must digest a large dose of modern mathematical logic and then move on to rather more esoteric topics. Among the subjects dealt with are: interpretations and structures, semantic implication, sentential logic, boolean algebra, variants of sentential logic, decidable theories, the compactness theorem, the Löwenheim-Skolem theorem, categoricity [[sic]], the Herbrand-Gentzen theorem, Craig's theorem, diagonal arguments and the paradoxes, Church's theorem, and the Gödel theorems on incompleteness and consistency. The particular methods used in this book reflect the recent interest in the algebraic properties of formal logical structures. The text includes numerous non-trivial exercises which are left to the student, and a short but well-chosen bibliography. Finally, one would think that the price being asked for this admittedly high-quality book is a bit high.—H. P. K.