Abstract

It is difficult to describe this book without praising it. Collected here in one volume are some thirty-six high quality translations into English of the most important foreign-language works in mathematical logic, as well as articles and letters by Whitehead, Russell, Norbert Weiner and Post. The contents of the volume are arranged in chronological order, beginning with Frege's Begriffsschrift—translated in its entirety—and concluding with Gödel's famous "On Formally Undecidable Propositions" and Herbrand's "On the Consistency of Arithmetic". The translation of the Gödel article is a small masterpiece, and is especially welcome in view of some of the rather poor efforts which have been made to render this important work into English. But this is more than a paste-pot job of anthologizing. Van Heijenoort and others have written prefaces to each of the included selections, in which are provided background historical and bibliographical information, comments on the intellectual currents which led to the production of the article, and, in many cases, a digest of the important theorems and concepts in the article itself. Each selection is a classic in its own right. In reading through the selections in this volume, one is struck by the extent to which purely mathematical problems and issues have controlled recent logical inquiry and have determined the development of various philosophies of mathematics. In the early stages, Frege and Peano were concerned with the logical reconstruction of arithmetic, and this led to the work of Russell and the logicist philosophy. Already in the development of the new logic there were certain problems which were resolved with Russell's theory of types—a device which many mathematicians felt to be rather artificial. The use of set theory, beginning with Cantor and reaching its maturity in Zermelo's 1904 proof of the well-ordering theorem, gave rise to a new set of difficulties concerning the axioms of choice, the continuum problem, and infinite sets. The two traditions are brought together in a common nexus of foundational problems by Weiner in 1914 and by Skolem's form of the Löwenheim theorem, given in 1920. Hereafter, the way is open for the rise of mathematical inquiry into the foundations of number systems, set theory and logic by Post, Skolem, von Neumann, Hilbert, Bernays, Gödel and many others. The work was far from completed by 1931, but a new and important tradition in mathematical inquiry had been established, and had come into its own. This book is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.—H. P. K.