Über das Unendliche. Offprint from: Mathematische Annalen 95. Bd., 2. Heft

HILBERT, David

Published by Springer, [Berlin, 1925

About this Item

THE HILBERT PROGRAM. First edition, very rare offprint, of this famous lecture which contains Hilbert's most detailed exposition of his proposal for the foundation of classical mathematics, which became known as Hilbert's Programme. "No one shall expel us from the paradise which Cantor has created for us," Hilbert famously declared in this lecture (p. 170). This paper further contains his attempted proof of the 'continuum hypothesis', that no set can have both 'fewer' elements than the set of real numbers and 'more' than the set of integers - the terms 'more' and 'fewer' referring to the (infinite) numbers of elements in these sets, understood in terms of Georg Cantor's theory of 'cardinal numbers' of infinite sets. "Hilbert devoted to the foundations of mathematics a series of papers . Among all these contributions the 1925 paper [the offered item] stands out as the most comprehensive presentation of Hilbert's ideas. It is the text of an address delivered in Münster on 4 June 1925 at a meeting organized by the Westphalian Mathematical Society to honor the memory of Weierstrass . The paper consists of two quite distinct parts. The first is a clear and forceful presentation of Hilbert's ideas at the time on the foundations of mathematics. It starts by recalling how Weiserstrass eliminated references to 'infinity' in analysis (Hilbert could have mentioned Cauchy and D'Alembert) and reviews the role played by the infinite in physics, set theory, and, when we deal with general propositions, logic. Leaning on the example of arithmetic, it introduces the distinction between finitary and ideal propositions, and it undertakes a simultaneous formalization of logic and arithmetic . The second part of the paper is a sketch of an attempted proof of the continuum hypothesis. Stated for the first time by Cantor at the very beginning of the development of set theory (1878), presented by Hilbert as Problem no. 1 in his famous list of unsolved mathematical problems (1900), the continuum problem for years resisted the efforts of mathematicians . The continuum hypothesis was proved by Gödel (1938) to be consistent with the customary axioms of set theory and by Cohen (1963) to be independent of these axioms . In his review of Gödel's second paper on the consistency of the continuum (1939), Bernays wrote: 'The whole Gödel reasoning may also be considered as a way of modifying the Hilbert project for a proof of the Cantor continuum hypothesis, as described in [Über das Unendliche, 1925], so as to make it practical and at the same time generalizable to higher powers'" (Van Heijenoort, From Frege to Gödel, pp. 367-92). OCLC locates just two copies, in Germany and the Netherlands, but we have located three others: one in the Paul Hertz collection of the University of Pittsburgh, and two in a private collection. Hilbert's work on the foundations of mathematics has its roots in his work on geometry of the 1890s, culminating in his influential Grundlagen der Geometrie (1899). Hilbert believed that the proper way to develop any scientific subject rigorously required an axiomatic approach, which would enable the theory to be developed independently of any need for intuition, and would facilitate an analysis of the logical relationships between the basic concepts and the axioms. Hilbert realized that the most important questions are the independence and the consistency of the axioms. For the axioms of geometry, consistency can be proved by providing an interpretation of the system in the real plane, and thus the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatization and a consistency proof. Hilbert provided such an axiomatization in Über den Zahlbegriff (1900), but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favoured way of providing a foundation for analysis in Dedekind's work relied on dubious ass. Seller Inventory # 3181

Bibliographic Details

Title: Über das Unendliche. Offprint from: ...

Publisher: Springer, [Berlin

Publication Date: 1925

Binding: Hardcover

Edition: First edition.