The Consistency of Number Theory Via Herbrand's Theorem

In this sequel to [7] the method of the consistency proof presented there is extended to provide a proof of the ω -consistency of the systems of number theory which were there shown consistent. This proof yields sharp bounds on the ordinal recursions required to establish the κ -consistency of these systems. The main technical innovation of this proof is the extension of what are essentially the methods of Ackermann [1] for handling finite sets of critical formulae of the first and second kinds to apply as well to sets of critical formulae in which the rank ordering is transfinite. The notation, definitions, and results of [7] will be presupposed throughout; we suggest the reader keep a copy of that paper at hand.