 Ackermann continued in section III of his 1925 at the very spot where Hilbert and Bernays had left off. His paper, submitted to Mathematische Annalen in March of 1924, and the corrective work he did in 1925 led to the conviction that the consistency of elementary arithmetic had been established. The corrective work had been done to address difficulties von Neumann had pointed out, but was not published by Ackermann; it was only presented in the second volume of Hilbert and Bernays 1939 (pp.  93–130).[6]  Von Neumann’s own proof theoretic investigations, submitted to Mathematische Zeitschrift in July 1925, were published under the title Zur Hilbertschen Beweistheorie in 1927. Hilbert’s 1928 Bologna Lecture prominently took Ackermann’s and von Neumann’s work as having established the consistency of elementary arithmetic, the proof making use only of finitist principles. Let F be a theory containing exclusively such principles, like primitive recursive arithmetic PRA; the principles of PRA consist of the Peano axioms for zero and successor, the defining equations for all primitive recursive functions (defined in  note 3),  and quantifier-free induction. Now the significance of a consistency proof in F can be articulated as follows: