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Towards a Re-Evaluation of Julius König's Contribution to Logic

Published online by Cambridge University Press:  15 January 2014

Miriam Franchella*
Affiliation:
Dipartimento Di Filosofia, Universitá Degli Studi, V. Festa Del Perdono 7, 20122 Milano, ItalyE-mail:[email protected]

Abstract

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going to discover them by analysing the content of the book.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

Bernays, P. (1935), Hilberts Untersuchungen über die Grundlagen der Arithmetik, D. Hilbert Gesammelte Abhandlungen, vol. 3, Springer, pp. 196216.Google Scholar
Boole, G. (1847), Mathematical Analysis of logic, Macmillan, Barclay and Macmillan / George Bell, Cambridge / London.Google Scholar
Brouwer, L.E.J. (1907), Over de grondslagen der wiskunde, Maas and van Suchtelen, Amsterdam, (English translation, On the Foundations of Mathematics, L.E.J. Brouwer Collected Works (A. Heyting editor), vol. 1 (1975), North-Holland, Amsterdam, pp. 11–101).Google Scholar
Brouwer, L.E.J. (1908), De onbetrouwbaarheid der logische principes, Tijdschrift voor Wijsbegeerte, vol. 2, pp. 152158, (English translation, The Unreliability of Logical Principles, L.E.J. Brouwer Collected Works (A. Heyting editor), vol. 1 (1975), North-Holland, Amsterdam, pp. 107–111).Google Scholar
Brouwer, L.E.J.(1912), Intuitionisme en formalisme; inaugurale rede, Clausen, Amsterdam, (English translation, Intuition and Formalism, L.E.J. Brouwer Collected Works (A. Heyting editor), vol. 1 (1975), North-Holland, Amsterdam, pp. 123–138).Google Scholar
Dedekind, R. (1888), Was sind und was sollen die Zahlen?, Vieweg, Wiesbaden.Google Scholar
Descartes, R. (1963), Œuvres et lettres, Pleiade, Paris.Google Scholar
Frege, G. (1879), Begriffsschrift, Nebert, Halle.Google Scholar
Frege, G. (1893), Grundgesetze der Arithmetik, Pohle, Jena.Google Scholar
Grassmann, H. (1861), Lehrbuch der Arithmetik für höhere Lehranstalten, Th. Chr. Fr. Enslin., Berlin.Google Scholar
Hallett, M. (1984), Cantorian Set Theory and Limitation of Size, Oxford University Press, Oxford.Google Scholar
Hilbert, D. (1905a), Über die Grundlagen der Logik und der Arithmetik, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 17. August 1904, Teubner, Leipzig, (English translation, On the Foundations of Logic and Arithmetic, From Frege to Gödel: a Source Book in Mathematical Logic (J. van Heijenoort, editor), 1976, Harvard University Press, Cambridge, pp. 130–138).Google Scholar
Hilbert, D. (1905b), Logische Principien des mathematischen Denkens, Vorlesungen ausgearbeitet von Ernst Hellinger.Google Scholar
Hilbert, D. (1922), Neubegründung der Mathematik. Erste Mitteilung, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, vol. 1, pp. 157177.CrossRefGoogle Scholar
Hilbert, D. (1926), Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161190, (English translation, On the Infinite, From Frege to Gödel: a Source Book in Mathematical Logic (J. van Heijenoort, editor), 1976, Harvard University Press, Cambridge, pp. 369–392).Google Scholar
Hilbert, D. (1972), Historisches Wörterbuch der Philosophie, vol. Bd II, Schwabe, Basel.Google Scholar
Kneale, W. and Kneale, M. (1962), The Development of Logic, Oxford University Press, Oxford.Google Scholar
König, J. (1905a), Zum Kontinuum-Problem, Mathematische Annalen, vol. 60, pp. 177180.CrossRefGoogle Scholar
König, J. (1905b), Über die Grundlagen der Mengenlehre und das Kontinuumproblem, Mathematische Annalen, vol. 61, pp. 156160.CrossRefGoogle Scholar
König, J. (1906), Sur la théorie des ensembles, Comptes rendus de l'Académie des Sciences de Paris, pp. 110112.Google Scholar
König, J. (1907), Über die Grundlagen der Mengenlehre und das Kontinuumproblem, Mathematische Annalen, vol. 63, pp. 217221.CrossRefGoogle Scholar
König, J. (1914), Neue Grundlagen der Logik, Arithmetik und Mengenlehre, Von Veit, Leipzig.CrossRefGoogle Scholar
MacColl, H. (1897), Symbolical Reasoning, Mind, vol. 6, pp. 493510.CrossRefGoogle Scholar
MacColl, H. (1906), Symbolic logic and Its Applications, Longmans, Green & Co., London, New York, Bombay.Google Scholar
Moore, G.H. (1978), The Origins of Zermelo's Axiomatization of Set Theory, Journal of Philosophical Logic, vol. 7, pp. 307329.CrossRefGoogle Scholar
Moore, G.H. (1982), Zermelo's Axiom of Choice: its Origins, Development, and Influence, Springer, New York.CrossRefGoogle Scholar
Neumann, J. von (1927), Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26, pp. 146.CrossRefGoogle Scholar
Peano, G. (1889), Arithmetices principia nova methodo exposita, Bocca & Clausen, Torino.Google Scholar
Peckhaus, V. (1990), Hilbertprogramm und kritische Philosophie, Vandenhoeck & Ruprecht, Göttingen.Google Scholar
Peckhaus, V. (1994), Logic in Transition: the Logical Calculi of Hilbert (1905) and Zermelo (1908), Logic and Philosophy of Science in Uppsala (Prawitz, D. and Westerstähl, D., editors), Kluwer, Amsterdam, pp. 311323.CrossRefGoogle Scholar
Peckhaus, V. (1996), The influence of Hermann Günther Grassmann and Robert Grassmann on Ernst, Schröder's Algebra of Logic, Hermann Günther Grassmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar (Schubring, G., editor), Kluwer, Dordrecht, Boston, London, pp. 217227.CrossRefGoogle Scholar
Poincaré, H. (1894), Sur la nature du raisonnement mathématique, Revue de métaphysique et de morale, vol. 2, pp. 371384.Google Scholar
Poincaré, H. (1902), La Science et l'Hypothèse., Bibliothèque de philosophie scientifique, Paris.Google Scholar
Poincaré, H. (1905), Les mathématiques et la logique, Revue de métaphysique et de morale, vol. 13, pp. 815835.Google Scholar
Poincaré, H. (1906), Les mathématiques et la logique, Revue de métaphysique et de morale, vol. 14, pp. 294317.Google Scholar
Poincaré, H. (1908), L'invention mathématique, vol. 10, pp. 357371.Google Scholar
Poincaré, H. (1909), La logique de l'infini, Revue de métaphysique et de morale, vol. 17, pp. 461482.Google Scholar
Post, E.L. (1921), Introduction to a General Theory of Elementary Propositions, American Journal of Mathematics, vol. 43, pp. 163185.CrossRefGoogle Scholar
Rescher, N. (1969), Many-valued Logic, McGraw-Hill, New York.Google Scholar
Russell, B. and Whitehead, A.N. (1910), Principia Mathematica I, Cambridge University Press, Cambridge.Google Scholar
Russell, B. and Whitehead, A.N. (1911), Principia Mathematica II, Cambridge University Press, Cambridge.Google Scholar
Russell, B. and Whitehead, A.N. (1912), Principia Mathematica III, Cambridge University Press, Cambridge.Google Scholar
Schröder, E. (1890), Vorlesungen über die Algebra der Logik, vol. 1, Teubner, Leipzig.Google Scholar
Sigwart, C. (1873), Logik, 3rd ed., Taupp, Tübingen, (1904), (This was the edition to which König referred).Google Scholar
Szénássy, B. (1992), History of Mathematics in Hungary until the 20th century, Springer, Berlin.CrossRefGoogle Scholar
Wang, H. (1957), The Axiomatization of Arithmetic, Journal for Symbolic Logic, vol. 22, pp. 145158.CrossRefGoogle Scholar
Wittgenstein, L. (1921), Logisch-philosophische Abhandlung, Annalen der Naturphilosophie, vol. 14, pp. 185262.Google Scholar
Zermelo, E. (1904), Beweis, dass jede Menge wohlgeordnet werden kann, Mathematische Zeitschrift, vol. 59, pp. 514516.CrossRefGoogle Scholar
Zermelo, E. (1908), Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65, pp. 261281.CrossRefGoogle Scholar