The built-in resources of LA– 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 LAcan express (‘express’ in the sense of Section 5.4). We now radically extend our list of examples by proving that LAcan in fact express any primitive recursive function.

Starting the proof

Our target, then, is to prove

The proof strategy Suppose that the following three propositions are all true:

1. LAcan express the initial functions.

2. If LAcan express the functions g and h, then it can also express a function f defined by composition from g and h.

3. If LAcan 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 LA, 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 LA.