 Bernays’s notes of the 1921/22 lectures reflect the consequence of that change in attitude. They contain a substantial development of “finitist arithmetic” and “finitist logic” in a quantifier-free formalism. Finitist arithmetic involves induction and primitive  recursion[3]  from the outset, and the central metamathematical arguments all proceed straightforwardly by induction on proof figures. The third part of these lectures is entitled The grounding of the consistency of arithmetic by Hilbert’s new proof theory. Here we find the first significant consistency proof—from a finitist  perspective.[4]  The proof is sketched in Hilbert’s Leipzig talk (Hilbert 1923: 184) and was presented in a modified form during the winter term of 1922/23; in that form the proof is given in Ackermann 1925: 4–7. Ackermann’s article was submitted for publication in early 1924, and by then the proof had taken on a certain canonical form that is still found in the presentation of Hilbert and Bernays 1934: 220–230. Let us see what was achieved by following Ackermann’s concise discussion.