Introduction

Abstract

The history of proof theory begins with the foundational crisis in the first decades of our century. At the turn of the century, as a reaction to the explosion of mathematical knowledge in the last two centuries, endeavours began to provide the growing body of mathematics with a firm foundation. Some of the notions used then seemed to be quite problematic This was especially true of those which somehow depended upon that of infinity. On the one hand there was the notion of infinitesimals which embodied ‘infinity in the small’. The elimination of infinitesimals by the introduction of limit processes represented a great progress in foundational work (although one may again find a justification for infinitesimals as it is done today in the field of nonstandard analysis). But on the other hand there were also notions which, at least implicitly, depended on ‘infinity in the large’. G.Cantor in his research about trigonometrical series was repeatedly confronted with such notions. This led him to develop a completely new mathematical theory of infinity, namely set theory. The main feature of set theory is the comprehension principle which allows to form collection of possibly infinitely many objects (of the mathematical universe) as a single object. Cantor called the objects of the mathematical universe ‘Mengen’ usually translated by’ sets’. Set theory, however, soon turned out to be a source of doubt itself. Since Cantor’s comprehension principle allows the collection of all sets x sharing an arbitrary property E(x) into the set {x: E(x)} one easily runs into contradictions.¹⁾

Cantor himself was well aware of the distinction between sets and other collections which may lead to contradictions. See his letter to Dedekind from 27.7.1899 [Purkert et al. 1987]