In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...) of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program. Gödel's work is generally taken to show that Hilbert's Program cannot be carried out. It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. (shrink)

Summary This article presents three hitherto unpublished letters by Frank Plumpton Ramsey on the foundations of mathematics with commentary. One of the letters was sent to Abraham Fraenkel and the other two letters to Heinrich Behmann. The transcription of the letters is preceded by an account that details the extent of Ramsey's known contacts with mathematical logicians on the Continent.

Zermelos Zeit in Göttingen (1897?1910) kann als wissenschaftlich fruchtbarste Periode in seiner Karriere angesehen werden. Gleichwohl stehen bisher Untersuchungen aus. die eine Einbettung von Zermelos Werk in den biographischen und sozialen Kontext ermöglichen Die vorliegende Studie will diese Lücke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfüllen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten über die Grundlagen der Mengenlehre; (2) die Vorgeschichte und näheren Umstände des 1907 an Zermelo verliehenen Lehrauftrages für mathematische Logik und verwandte Gegenstände. mit dem ein erster (...) Schritt zur Institutionalisierung dieses Faches als mathematischer Teildisziplin gemacht wurde. Beides wird in enger Verbindung zur ersten Phase des Hilbertschen Programms zur Grundlegung der Mathematik gesehen. Es wird aber auch gezeigt, daß für die Erteilung des Lehrauftrages neben diesen systematischen und forschungspolitischen Motiven auch persönliche und institutspolitische Rücksichten ausschlaggebend waren. Im Anhang werden Teile eines Briefwechsels zwischen Ernst Zermelo und dem Göttinger Philosophen Leonard Nelson ediert sowie die Anträge der Direktoren des Göttinger mathematisch-physikalischen Seminars auf Erteilung des Lehrauftrags und auf Ernennung Zermelos zum außerordentlichen Professor. Zermelo's Göttingen period (1897?1910) can be regarded as the most fruitful period in his scientific career. Nevertheless, up till now there have been no investigations which enable us to embed his work in its biographical and social contexts. This study tries to partially fill the gap, concentrating on two major themes: (I) the historical context of the development of Zermelo's early work on the foundations of set theory; (2) the prehistory and particulars of his lectureship for mathematical logic and related topics, this lectureship, which he was awarded in 1907, being the first step taken in Germany towards the establishment of this subject as a separate mathematical discipline. Both will be seen in close connection with the first phase of Hilbert's programme of founding mathematics. It is shown, however, that this lectureship was not only created to further the aims of Hilbert's research programme but that personal and institutional motives also played a role. In the appendices, parts of a correspondence between Ernst Zermelo and the Göttingen Philosopher Leonard Nelson are edited, as well as the applications of the directors of the Göttingen Mathematisch-physikalisches Seminar for awarding the lectureship to Zermelo in 1907, and for appointing Zermelo to an extraordinary professorship in 1910. (shrink)

In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in (...) transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction. (shrink)

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain "general consistency result" due to Bernays. An analysis of the form of this so-called "failed proof" (...) sheds further light on an interpretation of Hilbert's programme as an instrumentalist enterprise with the aim of showing that whenever a "real" proposition can be proved by ?ideal? means, it can also be proved by "real", finitary means. (shrink)

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...) University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the -calculus, on which Hilbert's axiom systemswere based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics'' throughthe -notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time. (shrink)