Abstract
Mathematics on the turn of the 19th century was characterized by the intense development on the one hand and by the appearance of some difficulties in its foundations on the other. Main controversy centured around the problem of the legitimacy of abstract objects. The works of K. Weierstrass have contributed to the clarification of the role of the infinite in calculus. Set theory founded and developed by G. Cantor promised to mathematics new heights of generality, clarity and rigor. Unfortunately paradoxes appeared. Some of them were known already to Cantor (e.g. the paradox of the set of all ordinals and the paradox of the set of all setsl) and they could be removed by appropriate modifications of set theory (cf. Cantor’s distinction between absolut unendliche or inkonsistente Vielheiten and konsistente Vielheiten,i.e. between classes and sets2). Frege’s attempt to realize the idea of the reduction of mathematics to logic (which was in fact a continuation of the idea of the arithmetization of analysis developed among others by Weierstrass) led to a really embarrasing contradiction discovered in Frege’s system by B. Russell and known today as Russell’s antinomy or as the antinomy of non-reflexive classes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography
Ackermann, W.: 1924–1925, ‘Begründung des tedium non datur mittels der Hilbertschen Theorie der Widerspruchsfreiheit’, Math. Annalen 93, 1–36.
Ackermann, W.: 1940, ‘Zur Widerspruchsfreiheit der Zahlentheorie’, Math. Annalen 117, 162–194.
Apt, K. R. and Marek, W.: 1974, ‘Second Order Arithmetic and Related Topics’, Annals of Math. Logic 6, 177–239.
Arai, T.: 1990, ‘Derivability Conditions on Rosser’s Provability Predicates’, Notre Dame Journal of Formal Logic (to appear).
Bernays, P.: 1967, ‘Hilbert David’. In: Edwards, P. (Ed.) Encyclopedia of Philosophy, vol. 3, Macmillan and Free Press, New York, pp. 496–504.
Bernays, P.: 1970, ‘On the Original Gentzen Consistency Proof for Number Theory’. In: Myhill, J. et al. (Eds.) Intuitionism and Proof Theory, North-Holland Publ. Comp., Amsterdam, pp. 409–417.
Cantor, Georg: 1932, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts mit erlaeuternden Anmerkungen sowie mit Ergaenzungen aus dem Briefwechsel Cantor-Dedekind, Hrsg. E. Zermelo, Verlag von Julius Springer, Berlin; reprinted by Springer-Verlag, Berlin-Heidelberg-New York 1980.
Detlefsen, M.: 1979, ‘On Interpreting Gödel’s Second Theorem’, Journal of Philosophical Logic 8, 297–313.
Detlefsen, Michael: 1986, Hilbert’s Program. An Essay on Mathematical Instrumentalism, D. Reidel Publ. Comp., Dordrecht-Boston-Lancaster-Tokyo.
Detlefsen, M.: 1990, ‘On the Alleged Refutation of Hilbert’s Program Using Gödel’s First Incompleteness Theorem’, Journal of Philosophical Logic 19, 343–377.
Drake, F.R.: 1989, ‘On the Foundations of Mathematics in 1987’. In: Ebbinghaus, H.-D. et al. (Eds.) Logic Colloquium ‘87, Elsevier Science Publishers, Amsterdam, pp. 11–25. Feferman, S.: 1960, ‘Arithmetization of Metamathematics in a General Setting’, Fundamenta Mathematicae 49, 35–92.
Feferman, S.: 1964–1968, S.: 1964–1968, ‘Systems of Predicative Analysis’, Part I: Journal of Symbolic Logic 29, (1964), 1–30; part II: Journal of Symbolic Logic 33, (1968), 193–220.
Feferman, S.: 1988, ‘Hilbert’s Program Revisited: Proof-Theoretical and Foundational Reductions’, Journal of Symbolic Logic 53, 364–384.
Friedman, H.: 1975, ‘Some Subsystems of Second Order Arithmetic and Their Use’, Proceedings of the International Congress of Mathematicians, Vancouver 1974, vol. 1, Canadian Mathematical Congress, pp. 235–242.
Friedman, H.: 1977, personal communication to L. Harrington.
Friedman, H.: 1981, ‘On the Necessary Use of Abstract Set Theory’, Advances in Mathematics 41, 209–280.
Gentzen, G.: 1936, ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’, Math. Annalen 112, 493–565
Gentzen, G.: 1938, ‘Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie’, Forschung zur Logik und zur Grundlegung der exakten Wissenschaften, New Series 4, 19–44.
Gentzen, G.: 1969, ‘On the Relation Between Intuitionistic and Classical Arithmetic’. In: Szabo, M. E. (Ed.) The Collected Papers of Gerhard Gentzen, North-Holland Publ. Comp., Amsterdam, pp. 53–67.
Gentzen, G.: 1974, ‘Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie’, Archiv fuer math. Logik und Grundlagenforschungen 16, 97–112. English translation in Szabo, M. E. (Ed.), The Collected Papers of Gerhard Gentzen, North-Holland Publ. Comp., Amsterdam 1969.
Gödel, K.: 1931, ‘Über formal unentscheidbare Sätze der ‘Principia Mathematica’ und verwandter Systeme. I’, Monatshefte fuer Mathematik und Physik 38, 173–198.
Gödel, K.: 1933, ‘Zur intuitionistischen Arithmetik und Zahlentheorie’, Ergebnisse eines Mathematischen Kolloquiums 4, 34–38.
Gödel, K.: 1946, ‘Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics, 1–4’. First published in Davis, M. (Ed.) The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, Hewlett, N.Y., pp. 84–88.
Gödel, K.: 1958, ‘Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes’, Dialectica 12, 280–287.
Goodstein, R. L.: 1944, ‘On the Restriced Ordinal Theorem’, Journal of Symbolic Logic 9, 33–41.
Guaspari, D. and Solovay, R.: 1974, ‘Rosser Sentences’, Annals of Mathematical Logic 16, 81–99.
Hilbert, D.: 1901, D.: 1901, ‘Mathematische Probleme’, Archiv der Mathematik und Physik 1, 4l-63 and 213–237. Reprinted in Hilbert, D. Gesammelte Abhandlungen, Berlin 1935.
Hilbert, D.: 1926, ‘Über das Unendliche’, Math. Annalen 95, 161–190. English translation: ‘On the Infinite’, in: van Heijenoort, J. (Ed.) From Frege to Goedel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967, pp. 367–392.
Hilbert, D.: 1927, ‘Die Grundlagen der Mathematik’, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universitaet 6, 65–85. Reprinted in Hilbert, D. Grundlagen der Geometrie, 7th edition, Leipzig 1930. English translation: ‘The Foundations of Mathematics’, in van Heijenoort, J. (Ed.) From Frege to Goedel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967, pp. 464–479.
Hilbert, D. and Bernays, R: 1934–1939, Grundlagen der Mathematik, Bd.I 1934, Bd.II 1939, Springer-Verlag, Berlin.
Jeroslov, R.: 1975, ‘Experimental Logics and OZ-Theories’, Journal of Philosophical Logic 4, 253–267.
Kirby, L. and Paris, J.: 1977, ‘Initial Segments of Models of Peano’s Axioms’. In Lachlan, A., Srebrny, M., and Zarach, A. (Eds.) Set Theory and Hierarchy Theory V (Bierutowice, Poland, 1976), LNM 619, Springer-Verlag, Berlin-Heidelberg-New York, pp. 211–226.
Kirby, L. and Paris, J.: 1982, ‘Accessible Independence Results for Peano Arithmetic’, Bull. London Math. Soc. 14, 285–293.
Kitcher, Ph.: 1976, ‘Hilbert’s Epistemology’, Philosophy of Science 43, 99–115.
Kreisel, G.: 1958, ‘Hilbert’s Programme’, Dialectica 12, 346–372. Revised, with Postscript in Benacerraf, P. and Putnam, H. (Eds.) Philosophy of Mathematics, Englewood Cliffs, New Jersey, Prentice-Hall 1964, pp. 157–180.
Kreisel, G.: 1968, ‘A Survey of Proof Theory’, Journal of Symbolic Logic 33, 321–388.
Kreisel, G.: 1976, ‘What Have We Learned from Hilbert’s Second Problem?’. In Browder, F. (Ed.) Mathematical Developments Arising from Hilbert Problems, Proceedings of the Symposia in Pure Mathematics 28, Providence, pp. 93–130.
Kreisel, G. and Takeuti, G.: 1974, ‘Formally Self-Referential Propositions for Cut-Free Analysis and Related Systems’, Dissertationes Mathematicae 118, 4–50.
Löb, M. H.: 1955, ‘Solution of a Problem of Leon Henkin’, Journal of Symbolic Logic 20, 115–118.
Murawski, R.: 1976–1977, ‘On Expandability of Models of Peano Arithmetic I-III’, Studia Logica 35, (1976), 409–419; 35, (1976), 421–431; 36 (1977), 181–188.
Murawski, R.: 1984a, ‘Expandability of Models of Arithmetic’. In: Wechsung, G. (Ed.) Proceedings of Frege Conference 1984, Akademie-Verlag, Berlin, pp. 87–93.
Murawski, R.: 1984b, ‘G. Cantora filozofia teorii mnogogci’ (G. Cantor’s philosophy of set theory), Studia Filozoficzne 11–12, 75–88.
Murawski, R.: 1987, ‘Generalizations and Strengthenings of Gödel’s Incompleteness Theorem’. In: Srzednicki, J. (Ed.) Initiatives in Logic, Martinus Nijhof Publishers, Dordrecht-BostonLancaster, pp. 84–100.
Paris, J.: 1978, ‘some Independence Results for Peano Arithmetic’, Journal of Symbolic Logic 43, 725–731.
Paris, J. and Harrington, L.: 1977, ‘A Mathematical Incompleteness in Peano Arithmetic’. In Barwise, J. (Ed.) Handbook of Mathematical Logic, North-Holland Pubi. Comp., Amsterdam, pp. 1133–1142.
Peckhaus, Volker: 1990, Hilbertprogramm and Kritische Philosophie. Das Goettinger Modell interdisziplinaerer Zusammenarbeit zwischen Mathematik and Philosophie, Vandenhoeck & Ruprecht, Göttingen.
Prawitz, D.: 1981, ‘Philosophical Aspects of Proof Theory’. In: Flgistad, G. (Ed.) Contemporary Philosophy. A New Survey, Martinus Nijhoff Publishers, The Hague-Boston-London, pp. 235–277.
Ramsey, F. R: 1929, ‘On a Problem of Formal Logic’, Proc. London Math. Soc. (2) 30, 264–286.
Reid, Constance: 1970, Hilbert, Springer-Verlag, Berlin-Heidelberg-New York.
Resnik, M. D.: 1974, ‘On the Philosophical Significance of Consistency Proofs’, Journal of Philosophical Logic 3, 133–147.
Rosser, J. B.: 1936, ‘Extensions of Some Theorems of Gödel and Church’, Journal of Symbolic Logic 1, 87–91.
Schütte, Kurt: 1977, Proof Theory, Springer-Verlag, Berlin-Heidelberg-New York.
Sieg, W.: 1985, ‘Fragments of Arithmetic’, Annals of Pure and Applied Logic 28, 33–71.
Sieg, W.: 1988, ‘Hilbert’s Program Sixty Years Later’, Journal of Symbolic Logic 53, 338–348.
Simpson, S. G.: 1985a, ‘Friedman’s Research on Subsystems of Second Order Arithmetic’. In Harrington, L. et al. (Eds.), Harvey Friedman’s Research in the Foundations of Mathematics, North-Holland Publ. Comp., Amsterdam, pp. 137–159.
Simpson, S. G.: 1985b, ‘Reverse Mathematics’, Proc. of Symposia in Pure Mathematics 42, 461–471.
Simpson, S. G.: 1987, ‘subsystems of Z2 and Reverse Mathematics’, Appendix to Takeuti, G. Proof Theory, North-Holland Publ. Comp., Amsterdam 1987, pp. 432–446.
Simpson, S. G.: 1988a, ‘Partial Realizations of Hilbert’s Program’, Journal of Symbolic Logic 53, 349–363.
Simpson, S. G.: 1988b, ‘Ordinal Numbers and the Hilbert’s Basis Theorem’, Journal of Symbolic Logic 53, 961–974.
Simpson, Stephen G.: Subsystems of Second Order Arithmetic (to appear).
Smorytiski, Craig: 1977, ‘The Incompleteness Theorems’. In Barwise, J. (Ed.) Handbook of Mathematical Logic, North-Holland Publ. Comp., Amsterdam, pp. 821–865.
Smorytíski, Craig: 1981, ‘Fifty Years of Self-Reference in Arithmetic’, Notre Dame Journal of Formal Logic 22, 357–374.
Smoryßski, Craig: 1985, Self-Reference and Modal Logic, Springer-Verlag, New York-BerlinHeidelberg-Tokyo.
Smoryfiski, C.: 1988, ‘Hilbert’s Programme’, CWI Quarterly 1, 3–59.
Tait, W. W.: 1981, ‘Finitism’, Journal of Philosophy 78, 524–546.
Takeuti, Gaisi: 1987, Proof Theory, North-Holland Publ. Comp., Amsterdam. van Heijenoort, J. (ed.): 1967, From Frege to Goedel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass.
Visser, A.: 1989, ‘Peano’s Smart Children: A Provability Logical Study of Systems with Built-In Consistency’, Notre Dame Journal of Formal Logic 30, 161–196.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Murawski, R. (1994). Hilbert’s Program: Incompleteness Theorems vs. Partial Realizations. In: Woleński, J. (eds) Philosophical Logic in Poland. Synthese Library, vol 228. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8273-5_8
Download citation
DOI: https://doi.org/10.1007/978-94-015-8273-5_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4276-7
Online ISBN: 978-94-015-8273-5
eBook Packages: Springer Book Archive