Abstract

This technical and sophisticated book has a misleading title. In only 140 pages it has chapters entitled: Truth Functions, Sentence Logic, Model Theory, Predicate Logic, Recursive Functions, Formalized Arithmetic, Free-Variable Arithmetic, and Axiomatic Set Theory. Goodstein says little about, let alone studies, the historical development of any of these topics. The history consists of a few references to some of the individuals who have played a role in the development of the field; but not always to the major figures. For instance, his chapter on model theory contains no discussion of the development of classical model theory. Only Herbrand is noted as a developer of model theory. However, even if these eight studies in mathematical logic do not tell much about the development of the topics, they still have considerable value. The first chapter on truth functions and the seventh on free-variable arithmetic are the most intriguing because they contain more of what is not usually presented in logic books. The first two chapters on sentential logic are not the standard textbook treatment. This really is a book for other working logicians and mathematicians. Quickly, Goodstein moves into treating truth functions for any number of truth values as functions in modulo arithmetic and investigates the topic of finding Sheffer functions, viz., a function from which all others for a given number of truth values can be obtained. In the second chapter, he is soon presenting Intuitionistic sentential logic and has some remarks about sentential logics between Intuitionistic logic and classical logic. He also presents sequent calculi. Since the first two chapters contain only 36 pages, this suggests that the book would be suitable as a text only for advanced and sophisticated students.