Maria Hämeen-Anttila and Jan von Plato, eds,Kurt Gödel: The Princeton Lectures on Intuitionism

Abstract

This book publishes for the first time notes from two notebooks of Gödel which formed the basis of a course on intuitionism Gödel delivered at Princeton in the spring of 1941. These notes contain by far the most detailed treatment by Gödel (and anyone prior to Luckhardt [1973] and Troelstra [1973]) of his now famous functional (‘Dialectica’) interpretation which was published by Gödel himself only much later in the very brief paper [Gödel, 1958]. The essence of the Princeton notes is close to that of Gödel’s lecture at Yale from April 15, 1941, which was posthumously published in [Gödel, 1995], pp. 189–200. (See [Troelstra, 1995] for insightful comments on these notes.) However, the notes for the course given at Princeton are much more detailed covering in total nine lectures and contain interesting additional material and insights into what Gödel aimed to achieve by his interpretation. After Gödel’s functional interpretation was made public first by Kreisel in [1959] and then by Gödel himself in the aforementioned paper [1958], it received some constant interest until the early 70s, but then this interest declined for a while as Gentzen-style methods were considered to be superior being ‘ordinal informative’ and in 1978 the so-called |$A$|-translation [Friedman, 1978; Dragalin, 1980] provided a simple proof of the closure of intuitionistic systems such as Heyting arithmetic (HA) under Markov’s rule which by that time was considered to be a main application of the Dialectica interpretation. Beginning in the 90s (starting with [Kohlenbach, 1992]), Gödel’s functional interpretation saw a revival as a major tool in the emerging proof mining paradigm which aims at extracting new effective data from prima facie noneffective proofs in many areas of mathematics, most prominently in (nonlinear) analysis (see, e.g., [Kohlenbach, 2008]). This led to the development of many variants and extensions of Gödel’s original interpretation and numerous applications to core mathematics. By now, Gödel’s Dialectica paper [1958] is his second most cited (both according to Zentralblatt and MathSciNet) work on mathematical logic next to his celebrated incompleteness paper. Given this massive renewed interest in the Dialectica interpretation in the past thirty years, the publication of Gödel’s extensive course notes on the subject is of course highly relevant although some of its content had been already commented on in [Troelstra, 1995] (while certain parts of the notes were only discovered in 2017 according to the preface1). As these notes were written at the time when Gödel conceived his interpretation and contain many comments for himself on how to present things best in the lectures, they provide an insight into the way Gödel worked allowing the reader to, in a sense, look over his shoulder while he is working. That Gödel is still struggling himself with the material is illustrated by the fact that in one lecture he gives a wrong definition for the interpretation of ‘|$\vee$|’ (as in the notes to his Yale lecture) which he then acknowledges as false in the next lecture presenting a second attempt which is the final solution as known today.