17 - Q is p.r. adequate

Summary

In Chapter 7, we defined a theory to be ‘sufficiently strong’ iff it captures all effectively decidable numerical properties. We later remarked that even Q turns out to be sufficiently strong. We can't show that yet, however, because we do not have to hand a general account of effective computability, and hence of effective decidability.

However, we do now know about a large class of effectively computable numerical functions, namely the primitive recursive ones; and we know about a correspondingly large class of the effectively decidable numerical properties, again the primitive recursive ones. And in this chapter, we take a big step towards showing that Q is indeed sufficiently strong, by showing that it can in fact capture all p.r. functions and properties. In a phrase, Q is p.r. adequate.

The idea of p.r. adequacy

(a) As we have said before, the whole aim of formalization is to systematize and regiment what we can already do. Since we can informally calculate the value of a p.r. function for a given input in an entirely mechanical way – ultimately by just repeating lots of school-arithmetic operations – then we will surely want to aim for a formal arithmetic which is able to track these informal calculations. So it seems that we will want a formal arithmetic T worth its keep to be able to calculate the values of p.r. functions for specific arguments. And it seems a very modest additional requirement that it can also recognize that those values are unique. Which motivates the following definition:

A theory T is p.r.adequate iff, for every p.r. function f, there is a corresponding φ that captures f in T.

It immediately follows (from the definition of a p.r. property and Theorem 16.1) that if T is p.r. adequate, then it also captures every p.r. property and relation.

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