Hilbert’s “Verunglückter Beweis”, the first epsilon theorem, and consisteny proofs’, History and Philosophy of Logic 25:79–94

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 Programme 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.