GÖDEL, Kurt. Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling. Herausgegeben von Karl Menger. Heft 1-7
Leipzig & Berlin: B.G. Teubner, 1931-1936. All first editions.

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 propositions; (4) During the 33rd session of the colloquium, November 7, 1931, the American visitor Parry introduced an axiom system for “analytic implication,” a concept of logical consequence entailing the unprovability of A -> B whenever B contains a propositional variable not occurring in A. Following Parry’s demonstration (via multi-valued truth tables) of this characteristic property, Gödel suggested that a completeness proof be sought for Parry’s axioms, while noting that the question whether Heyting’s propositional calculus could be realized using only finitely many truth values was then open. On p. 4 of this same issue, an article by Alt (“Zur Theorie der Krümmung”) mentions an unpublished suggestion by Gödel; (5) To Hahn’s question, “Can every independence proof for statements of the propositional calculus be carried out by means of finite multi-valued truth tables?” Gödel provides a negative answer. Specifically, using infinitely many truth values he demonstrates the independence of p ⊃ ~~p from the set of axioms p ⊃ p, (p ⊃ ~~q) ⊃ (p ⊃ q), and (~~p ⊃ ~~q) ⊃ (p ⊃ q), while showing that any finite realization of those axioms must also realize p ⊃ ~~q; (6 & 7) Gödel’s contributions to geometry have been overlooked by bibliographers. Both [6] and [7] formed part of the 42nd colloquium, held February 18, 1932. In the former, Gödel answers a question raised by Laura Klanfer at the 37th colloquium, December 2, 1931: he shows that whenever a quadruple of points in a metric space is isometric to four noncoplanar points of R3, the quadruple is isometric, under the geodesic metric, to four points on the surface of a sphere. (The corresponding result for the usual metric on R3 is trivial.) In the second paper Gödel reformulates Wald’s axiomatization of the betweenness concept as a theorem about triples of real numbers, assigning the triple of distances (ab, bc, ac) to a triple (a, b, c) in a given metric space. The theorem states that b lies between a and c in the sense of Menger if and only if (ab, bc, ac) lies in that part of the plane x + y = z for which each of the four quantities x, y, z, and (x + y - z)(x - y + z)(-x + y + z) is nonnegative; (8) Only two brief comments were published from the discussion with the above title held as the 51st colloquium, May 25, 1932. In translation, Gödel’s remark reads in full: “[Some]one should investigate the system of all those statements about fields that in normal form contain no existential prefixes. The concepts of point and line, which are definable by application of existential prefixes (e.g., a point is an element for which there exists no nonempty element that is a proper part of it), are undefinable in this more restricted system.”; (9) In this short but important paper, Gödel shows that although the intuitionistic propositional calculus is customarily regarded as a subsystem of the classical, by a different translation the reverse is true, not only for the propositional calculus but for arithmetic and number theory as well. (Independently and slightly later the same result was discovered by Gentzen and Bernays. Specifically, with each formula A of Herbrand’s system of arithmetic Gödel associates a translation A' in an extension of Heyting’s arithmetic, such that A' is intuitionistically provable whenever A is classically provable. Since it provides an intuitionistic consistency proof for classical arithmetic, Gödel’s translation gives classical mathematicians grounds for maintaining that insofar as arithmetic is concerned, intuitionistic qualms amount to “much ado about nothing”; for intuitionists, however, the issue is not so much consistency as it is matters of proper interpretation and methodology; (10) By formalizing the concept “p is provable” via a unary predicate Bp satisfying the axioms Bp -> p, Bp -> BBp, and Bp -> (B(p -> q) -> Bq), Gödel shows that Heyting’s propositional calculus can be given a natural classical interpretation. Specifically the intuitionistic notions ⌉p, p ⊃ q,p v qy and p ∧ q are to be interpreted by ~Bp, Bp -> Bq, Bp v Bq, and p*q; (11) Though considerably more accessible than [the original printing in Anzeiger der Akademie der Wissenschaften in Wien, 1932], this reprint has not been cited in earlier bibliographies. The text is identical to the original except for the addition of an opening clause attributing the question to Hahn, (12) This brief note, part of the 53rd colloquium, November 10, 1932, is devoted to proving that every one-to-one mapping of the real projective plane into itself that carries straight lines into straight lines is a collineation; (13) Gödel’s only joint paper, previously uncited. A single, mildly technical result is established, whose aim is to show that so-called “volume determinants” are appropriate for giving a coordinate-free characterization of Gaussian surfaces. The paper is intended as a contribution to Menger’s program for making precise, in a coordinate-free way, the assertion that Riemannian spaces behave locally like Euclidean spaces; (14) In this two-sentence remark, Gödel notes that a realistic analysis of demand in economics must take into account a firm's income, which depends upon the cost of production; (15) A short but important paper, largely overlooked until the advent of computational complexity theory. In terms of the latter, [1936a] provides an early example of a speed-up theorem, viz., that in formal number theory, passing to higher types (allowing sets of integers, sets of sets of integers, etc.) not only causes some previously unprovable statements to become provable, but also greatly reduces the length of the shortest proofs of some previous theorems.

Seven seperate issues, in their original printed wrappers; very fine and fresh, scarce in such good condition. 8vo: 232 x 152 mm. pp. 31,(1:blank); 38,(2:blank); 26; 45,(1:blank); 42.