# A Most Interesting, but Revoked Draft for Hilbert and Bernays' "Grundlagen der Mathematik" that never found its way into any publication, and 2 CV of Gisbert Hasenjaeger

## TL;DR

This paper investigates a previously unknown draft related to Hilbert and Bernays' foundational work in mathematics, revealing its connection to Hasenjaeger's contributions and the impact of proof theory developments on the text's revision.

## Contribution

It uncovers and analyzes a lost draft of Hilbert-Bernays' Grundlagen der Mathematik and documents the collaborative work of Hasenjaeger and Bernays on the second edition.

## Key findings

- Identified a candidate for the revoked draft in Bernays' legacy archive.
- Provided evidence linking the draft to Hasenjaeger and Bernays' revision efforts.
- Suggested the draft involved a complete reorganization of the first volume.

## Abstract

In 1934, in Bernays preface to the 1st edn. of the 1st vol. of Hilbert-Bernays "Grundlagen der Mathematik", a nearly completed draft of the the finally two-volume monograph is mentioned, which had to be revoked because of the completely changed situation in the area of proof theory after Herbrand and Goedels revolutionary results. Nothing at all seems to be known about this draft and its whereabouts. A third of a century later, Bernays preface to the 2nd edn. (1968) of the 1st vol. of Hilbert-Bernays mentions joint work of Hasenjaeger and Bernays on the second edition. Bernays states there that it became obvious that the integration of the many new results in the area of proof theory would have required a complete reorganization of the book, i.e. that the inclusion of the intermediately found new results in the area of proof theory turned out to be unobtainable by a revision, but would have required a complete reorganization of the entire textbook. We document that - even after the need for a complete reorganization had become obvious - this joint work went on to a considerable extent. Moreover, we document when Hasenjaeger stayed in Zurich to assist Bernays in the completion of the 2nd edn. In May 2017, we identified an incorrectly filed text in Bernays scientific legacy at the archive of the ETH Zurich as a candidate for the beginning of the revoked draft for the 1st edn. or of a revoked draft for the 2nd edn. In a partial presentation and careful investigation of this text we gather only some minor evidence that this text is the beginning of the nearly completed draft of the 1st edn., but ample evidence that this text is part of the work of Hasenjaeger and Bernays on the 2nd edn. We provide some evidence that this work has covered a complete reorganization of the entire 1st vol., including a completely new version of its last chapter on the iota.