## [Thirteen papers, all first editions, in Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling. Herausgegeben von Karl Menger. Heft 1-5].

Leipzig & Berlin: B.G. Teubner, 1931-33.

First editions, and a fine set in the original printed wrappers, of these rare proceedings to which Gödel contributed thirteen important papers and remarks on the foundations of logic and mathematics (see below for a complete annotated listing). The most important are perhaps ‘Über Vollständigkeit und Widerspruchsfreiheit’ (‘On completeness and consistency’) in Heft 3 and ‘Zur intuitionistischen Arithmetik und Zahlentheorie’ (‘On intuitionist arithmetic and number theory’) in Heft 4. Based on the lecture at the Colloquium required for his *Habilitation*, in the first paper Gödel presented a different approach to his epochal incompleteness theorem, published just a few months earlier in *Monatshefte für Mathematik*: instead of Russell’s theory of types, in the present version he used Peano’s axioms for the natural numbers; this soon became the standard approach. In the second paper, Gödel proved that intuitionist mathematics is no more certain, or more consistent, than ordinary mathematics. “By invitation, in October 1929 Gödel began attending Menger’s mathematics colloquium, which was modelled 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 co-editor 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” (*DSB* XVII: 350).

Working under Hans Hahn, Karl Menger (1902-85) received his PhD from the University of Vienna in 1924 and accepted a professorship there three years later. “During the academic year 1928/29, several students asked Menger to direct a Mathematical Colloquium, somewhat analogous to the philosophically motivated Vienna Circle ... This Colloquium, which met on alternate Tuesdays during semester time, had a flexible agenda including lectures by members or invited guests, reports on recent publications and discussion of unsolved problems. Menger kept a record of these meetings, which he published, regularly in November of the following year, under the title ‘Ergebnisse eines mathematischen Kolloquiums’...

“Gödel had entered the university in 1924, and Menger first met him as the youngest and most silent member of the Vienna Circle. In 1928, Gödel started working on Hilbert’s program for the foundation of mathematics, and in 1929 he succeeded in solving the first of four problems of Hilbert, proving in his PhD thesis (under Hans Hahn) that first order logic is complete: Any valid formula could be derived from the axioms.

“At that time Menger, who was greatly impressed by the Warsaw mathematicians, had invited Alfred Tarski to deliver three lectures at the Colloquium. Gödel, who had asked Menger to arrange a meeting with Tarski, soon took a hand in running the Colloquium and editing its Ergebnisse.

“Menger was visiting the USA [in 1931] when Gödel discovered the incompleteness theorem and used it to refute the remaining three of Hilbert’s conjectures. He learned by letter that Gödel had lectured in the Colloquium ‘On Completeness and Consistency’. This was the lecture required for Gödel’s habilitation. The paper required for the same procedure, ‘On the undecidability of certain propositions in the Principia Mathematica,’ had been published in Hahn’s ‘Monatshefte’. In his Colloquium lecture, Gödel presented a simpler approach. Instead of Russell’s theory of types, he used Peano’s axioms for the natural numbers. This soon became the standard approach …

“Menger was particularly fond of Gödel’s results on intuitionism. These vindicated his own tolerance principle. Specifically, Gödel proved that intuitionist mathematics is no more certain, or more consistent, than ordinary mathematics (‘Zur intuitionistischen Arithmetik und Zahlentheorie’, Heft 4) … Menger brought Oswald Veblen to the Colloquium when Gödel lectured on this result. Veblen, who had been primed by John von Neumann, was tremendously impressed by the talk and invited Gödel to the Institute for Advanced Study during its first full year of operation: A signal honour that proved a blessing in Gödel’s later years” (Karl Sigmund in *Selecta Mathematica*, pp. 14).

The Gödel papers contained in these five volumes are as follows, with summaries from the Annotated Bibliography of Gödel by John Dawson:

(1) Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, Heft 2, pp. 27-28. 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) Über Vollständigkeit und Widerspruchsfreiheit, Heft 3, pp. 12-13. 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) Eine Eigenschaft der Realisierungen des Aussagenkalküls, Heft 3, pp. 20-21. 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) Untitled remark following W. T. Parry Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation), Heft 4, p. 6. 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) Über Unabhängigkeitsbeweise im Aussagenkalkül, Heft 4, pp. 9-10. 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) Über die metrische Einbettbarkeit der Quadrupel des R^{3} in Kugelflächen, Heft 4, pp. 16-17;

(7) Über die Waldsche Axiomatik des Zwischenbegriffes, Heft 4, pp. 17-18. 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 R^{3} 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) Zur Axiomatik der elementargeometrischen Verknüpfungsrelationen, Heft 4, p. 34. 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) Zur intuitionistischen Arithmetik und Zahlentheorie, Heft 4, pp. 34-38. 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) Eine Interpretation des intuitionistischen Aussagenkalküls, Heft 4, pp. 39-40. 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) Reprint of Zum intuitionistischen Aussagenkalkül [*Anzeiger der Akademie der Wissenschaften in Wien*, vol. 69,1932, pp. 65-66], Heft 4, p. 40. Though considerably more accessible than [the original printing], 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) Bemerkung über projektive Abbildungen, Heft 5, p. 1. 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) Diskussion über koordinatenlose Differentialgeometrie (with K. Menger and A. Wald), Heft 5, pp. 25-26. 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.

Three further parts of the *Ergebnisse* were published (1934-37), but these contain only one additional paper by Gödel. They are rarely found, and were probably published in much smaller numbers than the first five parts owing to the political turmoil in Vienna which began with the failed Nazi coup in July 1934. The Vienna Colloquium ended when Menger moved to the US in 1937 to take up a position at the University of Notre Dame. There he reinstated the Colloquium; its proceedings were published as *Reports of a Mathematical Colloquium* (Notre Dame: University Press, 1939-48).

J. W. Dawson, The Published Work of Kurt Gödel: An Annotated Bibliography, *Notre Dame Journal of Formal Logic* 24 (1983), 255-84; K. Menger, *Selecta Mathematica*, Springer, 2002.

Five separate issues, in their original printed wrappers. 8vo (232 x 152 mm), pp. 31, [1, blank]; 38, [2, blank]; 26; 45, [1, blank]; 42.

Item #3707

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