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.

I shall begin by reviewing Hilbert's original statement of his program. After that I shall explicate the program in precise terms which, although more formal than Hilbert's, remain completely faithful to his original intention. This formal version of the program is definitively refuted by Gödel's Theorem. But the formal version also provides a context in which partial realizations can be studied in a precise and fruitful way. I shall use this context to discuss the modern developments which were alluded to above. In addition I shall explain how these developments are related to so-called ``reverse mathematics.'' Finally I shall rebut some possible objections to this research and to the claims which I make for it.