This is the point at which we bring proof and recursion together and begin to study connections between the computational complexity of recursive functions and the logical complexity of their formal termination or existence proofs. The rest of the book will largely be motivated by this theme, and will make repeated use of the basics laid out here and the proof-theoretic methods developed earlier. It should be stressed that by “computational complexity” we mean complexity “in the large” or “in theory”; not necessarily feasible or practical complexity. Feasibility is always desirable if one can achieve it, but the fact is that natural formal theories of even modest logical strength prove the termination of functions with enormous growth rate, way beyond the realm of practical computability. Since our aim is to unravel the computational constraints implicit in the logic of a given theory, we do not wish to have any prior bounds imposed on the levels of complexity allowed.

At the base of our hierarchy of theories lie ones with polynomially or at most exponentially bounded complexity, and these are studied in part 3 at the end of the book. The principal objects of study in this chapter are the elementary functions, which (i) will be characterized as those provably terminating in the theory IΔ0(exp) of bounded induction, and (ii) will be shown to be adequate for the arithmetization of syntax leading toGödel's theorems, a fact which most logicians believe but which rarely has received a complete treatment elsewhere.