7 - Sufficiently strong arithmetics

Summary

Theorem 6.3, our first shot at an incompleteness theorem, applies to sound theories. But we have already remarked in Section 1.2 that Gödel's arguments show that we don't need to assume soundness to prove incompleteness. In this chapter we see how to argue from consistency to incompleteness.

But if we are going to weaken one assumption (from soundness to mere consistency) we'll need to strengthen another assumption: we'll now consider theories that don't just express enough but which can capture, i.e. prove, enough.

Starting in Chapter 10, we'll begin examining various formal theories of arithmetic ‘from the bottom up’, in the sense of first setting down the axioms of the theories and then exploring what the different theories are capable of proving. For the moment, however, we are continuing to proceed the other way about. In the previous chapter, we considered theories that have sufficiently expressive languages, and so can express what we'd like any arithmetic to be able to express. Now we introduce the companion concept of a sufficiently strong theory, which is one that by definition can prove what we'd like any moderately competent theory of arithmetic to be able to prove about decidable properties of numbers. We then establish some easy but deep results about such theories.

The idea of a ‘sufficiently strong’ theory

Suppose that P is some effectively decidable property of numbers, i.e. one for which there is an algorithmic procedure for deciding, given a natural number n, whether n has property P or not.