20 - Arithmetization in more detail

Summary

In the last chapter we gave informal but hopefully entirely persuasive arguments that key numerical properties and relations that arise from the arithmetization of the syntax of PA – such as Term, Wff and Prf – are primitive recursive.

Gödel, as we said, gives rigorous proofs of such results (or rather, he proves the analogues for his particular formal system). He shows how to define a sequence of more and more complex functions and relations by composition and recursion, eventually leading up to a p.r. definition of Prf. Inevitably, this is a laborious job: Gödel does it with masterly economy and compression but, even so, it takes him forty-five steps of function-building to show that Prf is p.r.

We have in fact already traced some of the first steps in Section 14.8. We showed, in particular, that extracting exponents of prime factors – the key operation used in decoding Gödöl numbers – can be done by a p.r. function, exf. To follow Gödel further, we need to keep going in the same vein, defining ever more complex functions. What I propose to do in this chapter is to fill in the next few steps moderately carefully, and then indicate rather more briefly how the remainder go. This should be quite enough to give you a genuine feel for Gödel's demonstration and to indicate how it can be completed, without going into too much unnecessary detail.