Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling. Herausgegeben von Karl Menger. Heft 1-7 GÖDEL, Kurt

 

Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling. Herausgegeben von Karl Menger. Heft 1-7

GÖDEL, Kurt.

 

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An absolutely mint set, in the original wrappers, of these rare proceedings to which Gödel contributed fifteen important papers and remarks on the foundations of logic and mathematics. "By invitation, in October 1929 Gödel began attending Menger’s mathematics colloquium, which was modeled on the Vienna Circle. There in May 1930 he presented his dissertation results, which he had discussed with Alfred Tarski three months earlier, during the latter’s visit to Vienna. From 1932 to 1936 he published numerous short articles in the proceedings of that colloquium (including his only collaborative work) and was coeditor of seven of its volumes. Gödel attended the colloquium quite regularly and participated actively in many discussions, confining his comments to brief remarks that were always stated with the greatest precision." (D.S.B. XVII: 350). The papers are: (1) Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, vol. 2, pp. 27-28; (2) Über Vollständigkeit und Widerspruchsfreiheit, vol. 3, pp. 12-13; (3) Eine Eigenschaft der Realisierungen des Aussagenkalküls, vol. 3, pp. 20-21; (4) Untitled remark following W. T. Parry Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation), vol. 4, p. 6; (5) Über Unabhängigkeitsbeweise im Aussagenkalkül, vol. 4, pp. 9-10; (6) Über die metrische Einbettbarkeit der Quadrupel des R3 in Kugelflächen, vol. 4, pp. 16-17; (7) Über die Waldsche Axiomatik des Zwischenbegriffes, vol. 4), pp. 17-18; (8) Zur Axiomatik der elementargeometrischen Verknüpfungsrelationen, vol. 4, p. 34; (9) Zur intuitionistischen Arithmetik und Zahlentheorie, vol. 4, pp. 34-38; (10) Eine Interpretation des intuitionistischen Aussagenkalküls, vol. 4, pp. 39-40; (11) Reprint of Zum intuitionistischen Aussagenkalkül [Anzeiger der Akademie der Wissenschaften in Wien, vol. 69,1932, pp. 65-66], vol. 4, p. 40; (12) Bemerkung über projektive Abbildungen, vol. 5, p. 1; (13) Diskussion über koordinatenlose Differentialgeometrie (with K. Menger and A. Wald), vol. 5, pp. 25-26; (14) Über die Produktionsgleichungen der ökonomischen Wertlehre, vol. 7, p. 6; (15) Über die Länge von Beweisen, vol. 7, pp. 23-24. John Dawson in his Annotated Bibliography of Gödel has the following summaries of these papers: (1) This undated contribution was not presented to a regular meeting of the colloquium, but appeared among the Gesammelte Mitteilungen for 1929/30. In the context of the first-order predicate calculus without equality, Gödel describes an effective procedure for deciding whether or not a formula with prenex form (3x1.xn)(y1y2)(3z1.zn)A(xi,yi,zi) is satisfiable; the procedure is related to the method used in [his dissertation Die Vollstandigkeit der Axiome des logischen Funktionenkalküls] to establish the completeness theorem; (2) Closely related to [Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, 1931], [this paper] notes extensions of the incompleteness theorems to a wider class of formal systems. The system considered in [his 1931 paper] is based on Principia Mathematica and allows variables of all finite types. Here Gödel observes that any finitely-axiomatizable, omega-consistent formal system S with just substitution and implication (modus ponens) as rules of inference will possess undecidable propositions whenever S extends the theory Z of first-order Peano arithmetic plus the schema of definition by recursion; and indeed, that the same is true of infinite axiomatizations so long as the class of Gödel numbers of axioms, together with the relation of immediate consequence under the rules of inference, is definable and decidable in Z; (3) In answer to a question of Menger, Gödel shows that given an arbitrary realization of the axioms of the propositional calculus in a structure with operations interpreting the connectives ~ and ?, the elements of the structure can always be partitioned into two disjoint classes behaving exactly like the classes of true and false pro. N° de réf. du libraire 2713

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