The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory

Abstract

Hilbert’s program is one of the truly magnificent projects in the philosophy of mathematics. To carry out this program he founded a new discipline of mathematics, called “Beweistheorie”, which was to perform the task of laying to rest all worries about the foundations of mathematics once and for all1 by securing mathematics via an absolute proof of consistency. The failure of Hilbert’s finitist reduction program on account of Godel’s incompleteness results is often gleefully trumpeted. Modern logic, though, has shown that modifications of Hilbert’s program are remarkably resilient. These modifications can concern different parts of Hilbert’s two step program2 to validate infinitistic mathematics. The first kind maintains the goal of a finitistic consistency proof. Here, of course, Godel’s second incompleteness theorem is of utmost relevance in that only a fragment of infinitistic mathematics can be shown to be consistent. Fortunately, results in mathematical logic have led to the conclusion that this fragment encompasses a substantial chunk of scientifically applicable mathematics (cf. [18, 72]). This work bears on the question of the indispensability of set-theoretic foundations for mathematics. The second kind of modification gives more leeway to the methods allowed in the consistency proof. Such a step is already presaged in the work of the Hilbert school. Notably Bernays has called for a broadened or extended form of finitism (cf. [4]). Rather than a finistic consistency proof the objective here is to give a constructive and predicative consistency proof for a classical theory T in which large parts of infinitistic mathematics can be developed. In order to undertake such a study fruitfully one needs to point to a particular formalization of constructive predicative reasoning P , and then investigate whether P is sufficient to prove the consistency of T . The particular framework I shall be concerned with in this paper is an intuitionistic and predicative theory of types which was developed by Martin-Lof. He developed his type theory “with the philosophical motive of clarifying the syntax and semantics of intuitionistic mathematics” ([41]). It is intended to be a full scale system for formalizing intuitionistic mathematics. Owing to research in mathematical logic over the last 30 years the program of reverse mathematics and Feferman’s work have been especially instrumental here one can take a certain fragment of second order arithmetic to be the system T . It turns out that Martin-Lof’s type theory P