1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

Is it possible to create formal proofs of interesting mathematical theorems which are mechanically checked in every detail and yet are readable and even faithful to the best expositions of those results in the literature? This paper answers that question positively for theorems about decidable properties of nite automata. The exposition is from Hopcroft and Ullman's classic 1969 textbook Formal Languages and Their Relation to Automata. This paper describes a successful formalization which is faithful to that book. The requirement of being faithful to the book has unexpected consequences, namely that the underlying formal theory must include primitive notions of computability. This requirement makes a constructive formalization especially suitable. It also opens the possibility ofusingthe formal proofs to decide properties of automata. The paper shows how to do this. 1

notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 1887--1963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifier-free expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...