Abstract
Since antiquity the mind of man has continually been vexed by the question of the infinite. Again and again it has been emphatically denied that anything infinite could exist or that any infinity could be conceived by man: again and again thinkers have arisen who found this perfectly possible. Aristotle had taught that something complete and infinite was in no way possible. The philosophers of the Middle Ages, dominated on the one hand by the authority of Aristotle (whom they styled ‘the Philosopher’ without further qualification) but on the other hand and to at least the same extent by the authority of the Church, never tired of discussing the question how Aristotle’s doctrine of the impossibility of the infinite could be reconciled with the Church’s doctrine of the omnipotence of God. St. Thomas Aquinas, refining and rendering more precise Aristotle’s thesis taught that nothing infinite could be given, whereas not a few of the impressive succession of Schoolmen — particularly the nominalists — defended the opposite thesis.1 A partial result, at all events, was the development of an admirable degree of logical rigour, which was completely lost in succeeding centuries and in many points only regained in the critical mathematics of the nineteenth century.
First published in Alte Probleme — Neue Lösungen in den exakten Wissenschaften, Fünf Wiener Vortrage, Zweiter Zyklus, Leipzig and Vienna, 1934.
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Notes
Cf. P. Duhem, Études sur Léonard de Vinci, ceux qu’il a lus et ceux qui l’ont lu, Seconde série (Paris, 1909), IX Léonard de Vinci et les deux infinis.
[Principia I.xxvi].
[Letter XIX to Foucher, Philosophische Schriften (ed. Gerhardt) Vol. I, p. 416. This passage is quoted on the title-page of Bolzano, see note 5 below.]
[In a letter to Schumacher quoted by Fraenkel, Einleitung, p. 1 (see note 6 below). Letter of 12 July 1831, Werke Vol. 8, p. 216.]
Die Paradoxien des Unendlichen (Leipzig, 1950), new edition with notes by H. Hahn (Leipzig, 1920) [E. T. Paradoxes of the Infinite (London, 1950)].
G. Cantor’s fundamental articles are contained in Georg Cantor, Gesammelte Abhandlungen, ed. E. Zermelo (Berlin, 1932). This volume also contains a portrait of Cantor and a biography composed by A. Fraenkel. For further study of set theory A. Fraenkel’s Einleitung in die Mengenlehre3 (Berlin, 1928) is to be recommended. [E. T. of some of Cantor’s articles in G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, tr. P. E. B. Jourdain (London & Chicago, 1915); E. T. of Fraenkel asFoundations of Set Theory (Amsterdam, 1958).]
For sets, M, consisting of one or two members, the rule holds only if one adds to the proper partial sets the empty set (null-set) possessing no members and the set M itself.
For further reading on the continuum problem see W. Sierpiński, Hypothèse du continu (Monografje Matematyczne, Tom. IV (Warsaw & Lvov, 1934). [The problems discussed here and in the next note have been decisively advanced by K. Gödel and P. J. Cohen. The former showed, “The axiom of choice and Cantor’s generalized continuum hypothesis… are consistent with the other axioms of set theory if these axioms are consistent” (The Consistency of the Axiom of Choice &c, Princeton, N. J., 1940, Introduction). The latter showed “that CH [the continuum hypothesis] cannot be proved from ZF [Zermelo-Fraenkel set theory] (with AC [the axiom of choice] included), and that AC cannot be proved from ZF” (Set Theory and the Continuum Hypothesis, New York, 1966 Introduction to Chapter IV)].
For further reading on the axiom of choice see W. Sierpiński, L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’analyse. Bull. de l’Acad. des Sciences de Cracovie, Classes des sciences math, et nat., Série A, 1918, pp. 97–152;
W. Sierpiński, Leçons sur les nombres transfinis (Paris, Gauthier-Villars, 1928). Chap. VI. [See also note 8 above.]
In the series of lectures in aid of a memorial to Ludwig Boltzmann. This lecture was published as H. Hahn, Logik, Mathematik und Naturerkennen (Einheitswissenschaft, No. 2, Vienna 1933). [It will be published in E. T. with the other Einheitswissenschaft pamphlets in a future volume of this Collection.]
Krise und Neuaufbau in den exakten Wissenschaften (Leipzig & Vienna 1933): H. Hahn, “Die Krise der Anschauung” [E. T. Chapter VIII of the present volume.]
Ibid.: K. Menger, ‘Die neue Logik’ [E. T. Chapter I of his Selected Papers in Logic and Foundations, Didactics, Economics, (Dordrecht, Boston, and London, 1979) in this Collection.]
Cf. A. N. Whitehead and B. Russell, Principia Mathematica 2 (Cambridge, 1925) Vol. I pp. 37ff.
Krise und Neuaufbau in den exakten Wissenschaften: G. Nöbeling, ‘Die vierte Dimension und der krumme Raum’.
H. Poincaré, La Science et l’Hypothèse (Paris, 1902) [E. T. Science and Hypothesis (London, 1905)].
For what follows see the easily intelligible presentation of A. Haas, Kosmologische Probleme der Physik (Leipzig, 1934) and the detailed but much more difficult one by H. Weyl, Raum, Zeit, Materie 5 (Berlin, 1923) [E. T. Space, Time, Matter (London, 1922)].
We are thus led to mathematical spaces that are three-dimensional analogues of a paraboloid of rotation. Cf. H. Weyl, Raum, Zeit, Materie 5 (Berlin, 1923) p. 257.
The following interpretation is due to E. A. Milne. Cf. A. Haas, Kosmologische Probleme der Physik (Leipzig, 1934) , p. 59
and E. Freundlich, Die Naturwissenschaften 21 (1933) p. 54.
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Hahn, H. (1980). Does the Infinite Exist?. In: McGuinness, B. (eds) Empiricism, Logic and Mathematics. Vienna Circle Collection, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8982-5_8
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