## Thèses présentées à la Faculté des Sciences de Paris pour obtenir le grade de Docteur ès Sciences Mathématiques ... 1re thèse: Recherches sur la théorie de la démonstration. 2me thèse: Propositions données par la faculté. Soutenues le 1930 devant la commission d'examen.

Warsaw: Dziewulski, 1930.

A fine copy of his doctoral dissertation - a seminal work in mathematical logic (Van Heijenoort: From Frege to Gödel). “Herbrand gave early signs of his mathematical gifts, entering the École Normale Supérieure at the exceptional age of seventeen and ranking first in the entering class. He completed his doctoral dissertation in April 1929. That October he began a year of service in the French army. He then went to Germany on a Rockefeller fellowship, studying in Berlin (until May 1931) with John von Neumann, then in Hamburg (May-June) with Emil Artin, and in Göttingen (June-July) with Emmy Noether. He left Göttingen for a vacation in the Alps and a few days later was killed in a fall at the age of twenty-three. ... Herbrand’s main contribution to logic was what is now called the Herbrand theorem, published in his doctoral dissertation: it is the most fundamental result in quantification theory. Consider an arbitrary formula F of quantification theory, then delete all its quantifiers and replace the variables thus made free with constants selected according to a definite procedure. A lexical instance of F is thus obtained. Let Fi be the ith lexical instance of F: the instances being generated in some definite order. The Herbrand theorem states the F is provable in anyone of the (equivalent) systems of quantification theory if and only if for some number K the disjunction F1 V F2 V ... V Fk (now called the kth Herbrand disjunction) is sententially valid ... The Herbrand theorem establishes an unexpected bridge between quantification theory and sentential logic. Testing a formula for sentential validity is a purely mechanical operation. Given a formula F of quantification theory, one tests the kth Herbrand disjunction of F successively for k=1, k=2, and so on: if F is provable, one eventually reaches a number k for which the kth Herbrand disjunction is valid. If F is not provable, there is, of course, no such k,. and one never learns that there is no such k (in accordance with the fact that there is no decision procedure for quantification theory). Besides yielding a very convenient proof procedure, the Herbrand theorem has many applications (a field explored by Herbrand himself) to decision and reduction problems and to proofs of consistency. Almost all methods for proving theorems by machine rest upon the Herbrand theorem” (D.S.B. Vol. 6, p.297). See also Hook & Norman: Origins of Cyberspace, no. 865 for the important application by John A. Robinson of Herbrand’s theorem. Van Heijenoort: From Frege to Gödel, pp. 525-581.

8vo: 239 x 169 mm. Original printed wrappers, two small pinholes in upper cover; an excellent copy. (2), 128, (2) pp. Custom cloth folding case with gilt spine lettering. Scarce.

Item #2294

Price: \$3,500.00

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