Logic in Transition: The Logical Calculi of Hilbert (1905) and Zermelo (1908)

Abstract

In lecture courses David Hilbert in 1905 and Ernst Zermelo in 1908 presented logical calculi which can be regarded as typical representatives of logical systems at a time when logic was in transition towards forming a new base for the foundations of mathematics. These calculi are the first fruits of discussions in David Hilbert’s circle in Göttingen which were provoked by the publication of the logical paradoxes by Bertrand Russell and Gottlob Frege in 1903. In the course of these discussions the Göttingen mathematicians in Hilbert’s circle reconsidered the interrelations between logic and mathematics, and fully grasped the eminent role of set theory for the foundation of mathematics. In this paper I intend to give a brief presentation of these calculi using hitherto unpublished material from the Nachltsse of Hilbert in Göttingen1 and Zermelo in Freiburg i.Br.2