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Partial realizations of Hilbert's program

  • Stephen G. Simpson (a1)

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.

I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.

Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

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[3] D. K. Brown and S. G. Simpson , Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123144.

[8] H. Friedman , S. G. Simpson and R. L. Smith , Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141181; addendum, vol. 28 (1985), pp. 319–320.

[10] K. Gödel , Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.

[15] D. Hilbert and P. Bernays , Grundlagen der Mathematik. Vols. I, II, 2nd ed., Springer-Verlag, Berlin, 1968, 1970.

[16] P. Kitcher . Hubert's epistemology, Philosophy of Science, vol. 43 (1976), pp. 99115.

[18] J. Lear , Aristotelian infinity, Proceedings of the Aristotelian Society (New Series), vol. 80 (1980), pp. 187210.

[20] W. Sieg , Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 3371.

[22] S. G. Simpson , Friedman's research on subsystems of second order arithmetic, Harvey Friedman's research in the foundations of mathematics ( L. Harrington et al., editors), North-Holland, Amsterdam, 1985, pp. 137159.

[25] W. W. Tait , Finitism, Journal of Philosophy, vol. 78 (1981), pp. 524546.

[28] E. P. Wigner , The unreasonable effectiveness of mathematics in the natural sciences, Communications on Pure and Applied Mathematics, vol. 13 (1960), pp. 114.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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