Hilbert’s “Verunglückter Beweis,” the first epsilon theorem, and consistency proofs

Zach, Richard. 2004. “Hilbert’s ‘Verunglückter Beweis’, the First Epsilon Theorem, and Consistency Proofs.” History and Philosophy of Logic 25 (2): 79–94. https://doi.org/10.1080/01445340310001606930.

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert’s Programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert’s epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain ‘general consistency result’ due to Bernays. An analysis of the form of this so-called ‘failed proof’ sheds further light on an interpretation of Hilbert’s Program as an instrumentalist enterprise with the aim of showing that whenever a ‘real’ proposition can be proved by ‘ideal’ means, it can also be proved by ‘real’, finitary means.

Review: Dirk Schlimm (Bulletin of Symbolic Logic 11/2 (2005) 247–248)