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Grundlagen der Mathematik II (Grundlehren der mathematischen Wissenschaften) (German Edition), by David Hilbert, Paul Bernays
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Der vorliegende Band schließt die Darstellung der Beweistheorie ab, die ich vor einigen Jahren zusammen mit P. BERNAYS begann. Auf meinen Wunsch hat P. BERNAYS wieder die Abfassung des Textes über nommen. Ich danke ihm für die Sorgfalt und Treue, mit der er meine Gedanken wiedergegeben hat, an deren Entwicklung er in jahrelanger Zusammenarbeit aufs stärkste beteiligt war. Ohne seine Mithilfe wäre die Vollendung dieses Buches unmöglich gewesen. Den Herren W. ACKERMANN, G. GENTZEN, A. SCHMIDT, H. SCHOLZ danke ich für ihre freundliche Mitwirkung bei den Korrekturen. Göttingen, im März 1939 HILBERT Zur Einführung Das vorliegende Buch soll einer eingehenden Orientierung über den gegenwärtigen Stoff der HILBERTschen Beweistheorie dienen. Wenn gleich das bisher hier Erreichte gemessen an den Zielen der Theorie sehr bescheiden ist, so liegt doch ein reichlicher Stoff an prägnanten Ergebnissen, an Gesichtspunkten und Beweisgedanken vor, die zur Kenntnis zu bringen als lohnend erscheint. Für die inhaltliche Gestaltung dieses zweiten Bandes waren durch den Zweck des Buches zwei Hauptthemata vorgezeichnet. - Es handelte sich einmal darum, die hauptsächlichen, an das e-Symbol sich knüpfenden beweistheoretischen Ansätze HILBERTS und ihre Durchführung zur ein gehenden Darstellung zu bringen.
- Sales Rank: #11662850 in Books
- Published on: 1970-01-01
- Released on: 1970-01-01
- Original language: German
- Number of items: 1
- Dimensions: 9.25" h x 1.32" w x 6.10" l, 1.78 pounds
- Binding: Paperback
- 568 pages
Most helpful customer reviews
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Historically important attempt to prove consistency of mathematics.
By Alan U. Kennington
I bought this from the German Amazon web site 6 months ago because I couldn't get it on the US Amazon web site at that time. It has enormous historical importance, and is cited by most books on this area of the foundations of mathematics. Both Volumes I and II were published at turning points in the history of mathematics. So I am glad to see that they are now available on Amazon USA.
Volume I of Grundlagen der Mathematik was published in 1934. The preface to Volume I in March 1934 stated that because of the publications of Herbrand and Gödel, it would be necessary to split the "Grundlagen" into two volumes. The issue was basically Gödel's 1931 famous incompleteness theorem, which was perceived as throwing a spanner in the works of Hilbert's formalist program to prove the consistency of mathematics. Therefore the existence of Volume II is a direct consequence of Gödel's incompleteness theorem for number theory.
The content of this 1939 Volume II of Grundlagen der Mathematik is mostly directed towards proving the consistency of arithmetic despite Gödel's theorem! It turns out that Gentzen in 1936/38 showed that arithmetic is after all consistent. Gentzen used some kind of triple induction equivalent to transfinite induction up to the ordinal number ε_0 (which by the way is a countable ordinal number). Kleene in 1963, in "Mathematical Logic", page 478, wrote the following comment.
"Gentzen's discovery is that the Gödel obstacle to proving the consistency of number theory can be overcome by using transfinite induction up to a sufficiently great ordinal."
This observation is the subject of pages 372–387 of this Volume II, where Hilbert and Bernays give a slightly modified version of Gentzen's proof. The whole of Volume II is pretty much concerned with this objective, which was delayed for 5 years by Gödel's theorem!
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