LA can express every p.r. function

Abstract

The built-in resources of L A – the first-order language of basic arithmetic which we first introduced in Section 5.2 – are minimal: there are no non-logical predicates, and just three primitive functions, successor, addition and multiplication. We have previously noted, though, a few examples of what else L A can express (‘express’ in the sense of Section 5.4). We now radically extend our list of examples by proving that L A can in fact express any primitive recursive function. Starting the proof Our target, then, is to prove Theorem 15.1 Every p.r. function can be expressed in L A . The proof strategy Suppose that the following three propositions are all true: L A can express the initial functions. If L A can express the functions g and h , then it can also express a function f defined by composition from g and h . If L A can express the functions g and h , then it can also express a function f defined by primitive recursion from g and h . Now, any p.r. function f must be specifiable by a chain of definitions by composition and/or primitive recursion, building up from initial functions. So as we follow through the full chain of definitions which specifies f, we start with initial functions which are expressible in L A , by (1). By (2) and (3), each successive definitional move takes us from expressible functions to expressible functions. So, given (1) to (3) are true, f must therefore also be expressible in L A .