Introduction

Abstract

Logic calculi can serve for many purposes, such as reconstructing and analyzing mathematical proofs in a formal manner or automating the search for mathematical proofs. As the paradoxes of set theory struck the mathematical community around 1900, the formal and consistent representation of theories became a central issue in foundational research. In the theory of proofs fascinating results were obtained in the 1930s, which culminated in the completeness and incompleteness (or better “incompletability”) results of Godel [Göd30],[Göd31]. Somewhat later Gentzen defined a natural notion of formal proofs [Gen34], which is closer to actual mathematical deduction than the so-called Hilbert-type systems. Like Herbrand [Her30] Gentzen investi­gated the form and structure of mathematical proofs, while Gödel’s famous theorems were more directed to provability.