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Partial realizations of Hilbert's program

  • Stephen G. Simpson (a1)
Extract

§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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[1]Bernays P., Hilbert, David, Encyclopedia of Philosophy (Edwards P., editor), Vol. 3, Macmillan and Free Press, New York, 1967, pp. 496504.
[2]Brown D. K., Functional analysis in weak subsystems of second-order arithmetic, Ph.D. Thesis, Pennsylvania State University, University Park, Pa., 1987.
[3]Brown D. K. and Simpson S. G., Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123144.
[4]Corcoran J., Review of [17], Mathematical Reviews 82c:03013.
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[6]Friedman H., Systems of second order arithmetic with restricted induction. I, II (abstracts), this Journal, vol. 41 (1976), pp. 557559.
[7]Friedman H., personal communication to L. Harrington, 1977.
[8]Friedman H., Simpson S. G. and Smith R. L., Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141181; addendum, vol. 28 (1985), pp. 319–320.
[9]Gödel K., On formally undecidable propositions of Principia Mathematica and related systems. I, translated by van Heijenoort J., [27], pp. 596616.
[10]Gödel K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.
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[21]Simpson S. G., Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?, this Journal, vol. 49 (1984), pp. 783802.
[22]Simpson S. G., Friedman's research on subsystems of second order arithmetic, Harvey Friedman's research in the foundations of mathematics (Harrington L.et al., editors), North-Holland, Amsterdam, 1985, pp. 137159.
[23]Simpson S. G., Subsystems of Z2 and reverse mathematics, appendix to G. Takeuti, Proof theory, 2nd ed., North-Holland, Amsterdam, 1987, pp. 432446.
[24]Simpson S. G., Subsystems of second order arithmetic (in preparation).
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[27]Van Heijenoort J. (editor), From Frege to Godel: a source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967.
[28]Wigner E. P., The unreasonable effectiveness of mathematics in the natural sciences, Communications on Pure and Applied Mathematics, vol. 13 (1960), pp. 114.
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