Abstract
Soon after assuming my position at the University of Vienna in the fall of 1927, I offered a quite well attended one-semester course which was on dimension theory. The name of one of the students who had enrolled was Kurt Gödel. He was a slim, unusually quiet young man. I do not recall speaking with him at that time.
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Notes
Jahresbericht d. deutschen Mathematikervereinigung 37 (1928) 213–226, and 298-308. Cf. also my book Selected Papers on Logic and Foundations, Didactics, Economics, Dordrecht 1979, quoted subsequently as Selected Papers, Chapter V.
For the development of the logical Principle of Tolerance cf. Selected Papersi pp. 11-16.
Erkenntnis 2, (1931), 147f.
Ergebnisse eines mathematischen Kolloquiums, 3 (1931-32) 12f. In the fall of 1930, a summary of Gödel’s fundamental results appeared in the Anzeiger of the Austrian Academy of Science, Nr. 19. His famous paper ‘Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme, I’ (On formally undecidable propositions of Principia mathematics and related systems I) was published in Monat-shefte für Mathematik, 38 (1931) 173-198, (also Kurt Gödel Collected Works Oxford and New York, 1986, hereafter CW, pp. 141-195) see also the short summary in Erkenntnis 2, 1931 149f.
Ergebnisse 3, 20f. (CW, pp. 238-41) Without any appeal to true and false the formalism of the traditional propositional calculus can be based on the following assumption: Let be the set of undefined elements (called propositions) with: 1) two operations: one unitary, denoted by ∼ (read not), and one binary denoted by ⊃ (read implies): 2) a nonempty subset ℑ (whose elements are called tautologies) satisfying the following conditions: a) if p and p ⊃ q belong to ℑ, then so does q; β) the axioms of the traditional propositional calculus, e.g. those of Lukasiewicz: (∼ p ⊃ p) ⊃ p, p ⊃ (∼ p⊃ q), (p⊃ q) ⊃ [ (q⊃ r) ⊃ (p ⊃ r)] are elements of ℑ. Under this assumption, Gödel proves that is the union of two disjoint subsets and s(whose elements may be called true and false, repectively), with the following properties: a) for each pair of propositions p, ∼ p, one belongs to and the other to: b) the proposition p⊃ q belongs to if and only if p belongs to and q to Calling a set with two operators, ∼ and⊃, and a subsetℑ satisfying the assumptions 1) and 2) a model of the traditional propositional calculus one can formulate the preceding theorem as follows: The elements of each model of the traditional calculus of propositions fall into two classes whose elements behave exactly like the true and the false propositions of that calculus. I should like to add here that my original question to Gödel may be extended to multi-valued propositional calculi.
‘Die neue Logik’ in the booklet quoted in 7 pp. 94-122. Translated as The New Logic’ in Philosophy of Science 4, 299-336, and reprinted in Selected Papers, pp 17-45.
Krise und Neuaufbau in den exakten Wissenschaften, Leipzig und Vienna, 1933, 122pp.
Review of Scientific Instruments 4 (1933). Cf. Selected Papers, p. 17f.
Gödel mentions three propositions satisfied in each model such that p⊃∼∼p, while there exists an infinite model not satisfying them.
C. I. Lewis’ system had been extensively discussed by Dr. W. T. Parry (4, 15f). In a discussion after an earlier paper by Dr. Parry, Gödel had proposed the following interpretation of Parry’s ‘p implies q analytically’: ‘q can be derived from (ist ableitbar aus) p and the axioms of logic, and contains no other concepts than p’ (4,6, CW pp. 266-7).
Math. Annalen 100 (1928) 77f.
Math. Annalen 104 (1931) 476–484.
The W is characterized among the subsets of (*) by the following properties: If the point (x,y,z) belongs to W, then so does (y,x,z), but not (z,y,x). If (u,v,w) and (w,x,y) belong to W, then so do the points (u,x,y) and (v,x,z) for every z provided that they belong to (*). For each z >, the set W contains at least one point (x,y,z). The set consisting of (0,z,z), (z,0,z) and all points (x,y,z) in W is closed.
Jahresber. DMV 37 (1928), Annals of Math. 37 (1936) 456-482.
Cf. Ergebnisse 1, 28f.; 4,14f, and 34,7; 11f, and Annals of Math. loc. cit. 14.
Math. Annalen 103 (1930) 466–501.
The radius of the circumcircie of three points is defined metrically as the quotient of the product of the three sides by the area of the triangle (expressed by Hero’s formula).
‘An example of a new type of cosmological solutions of Einstein’s field equations of gravitation.’ Review of Modern Physics 21, 1949, pp. 555–62.
Wissenschaftliche Weltauffassung: Der Wiener Kreis, Wien 1929. E. T. in O. Neurathl Empiricism and Sociology, Vienna Circle Collection, Dordrecht.
Elements d‘Economie Politique Pure, 2 ed. Lausanne 1889.
Ergebnisse 6 10 f.
Ergebnisse 6, 12–18.
Ergebnisse 7, 1–6.
Ergebnisse 7, 6.
Earlier_that year, Godel participated in a discussion by Tarski and myself on Friedrich Waismann’s Bemerkungen zu Freges und Russells Definition der Zahl, (Remarks on Frege’s and Russell’s definition of numbers) no details of which have been recorded. [ Waismann’s paper was no doubt identical in content with the relevant section of his Einführung in das Mathemathische Denken (E. T. Introduction to Mathematical Thinking) Vienna 1936 (New York, 1951) B. McG.]
Ergebnisse 7, 23, CW pp. 396–399.
About that period, cf. my ‘Memories of Moritz Schlick’ in Rationality and Science ed. E. T. Gadol, Vienna 1982.
I do not recall any instance of his discussing socio-economic questions.
A three-day meeting on all aspects of the calculus of variations.
A resort in Styria.
Alt and Wald arranged a couple of meetings after my departure, as they had promised me; but no minutes were kept.
A three day meeting on algebra of geometry and other aspects of lattice theory. No proceedings were published.
Cf. papers by F. P. Jenks, J.C. Abbott, B. T. Topel, J. Landin in the first 6 issues of Reports of a Mathematical Colloquium, 2nd Series, Notre Dame, 1938-45.
‘Russell’s mathematical logic’ in The Philosophy of Bertrand Russell, Library of Living Philosophers, ed. P. A. Schilpp, Evanston 1944, 123–153.
[Remark by Eckehart Köhler: In June 1985, shortly before his death in October 1985, Karl Menger expressed to me in a telephone call that he had only recently learned of the fact that Gödel had married Adele in September 1938, and that had he known that in 1939-40, his attitude to Gödel’s return then and subsequently would have been completely different; and that the corresponding passages of this memoir on Gödel should be amended.]
Amer. Math. Monthly, 54 (1947) 515–525.
Principles of Mathematics, Cambridge 1903, p. 5 and p. 89.
AII these matters are extensively treated in my Selected Papers, Parts III and IV. The clarifications are of course also relevant for the didactics of pure and applied analysis and of mathematical science. But this is a field in which Gödel (in contrast to his occasional interest in logical didactics) had no experience and took only little interest.
On the other hand, J. von Neumann told me (and our common friend Morgenstern) that after thinking over the arguments presented above he became aware of previously unnoticed obscurities in the mode of expression in our mathematico-scientific discourse.
Friedrich Waismann, Wittgenstein und der Wiener Kreis, ed. B. McGuinness, Oxford 1967. (E.T. Wittgenstein and the Vienna Circle, Oxford, 1979)
[In all subsequent editions this is the third appendix to Part I: the passage referred to occurs at the end of the penultimate paragraph. The word “logische”/ “logical” does not occur in Wittgenstein’s text, though the word “Kunststucken”/ “conjuring tricks” or “legerdomain” does. B. McG.]
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Menger, K. (1994). Memories of Kurt Gödel. In: Golland, L., McGuinness, B., Sklar, A. (eds) Reminiscences of the Vienna Circle and the Mathematical Colloquium. Vienna Circle Collection, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1102-7_17
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