18 - Interlude: About the First Theorem

Summary

We have achieved our first main goal, namely to prove Gödel's First Incompleteness Theorem. And it will do no harm to pause for breath and quickly survey what we've established and how we established it. Equally importantly, we should make it clear what we have not proved. The Theorem attracts serious misunderstandings. We will briefly block a few of these.

What we've proved

To begin with the headlines about what we have proved (we are going to be repeating ourselves, but – let's hope! – in a good way). Suppose we are trying to regiment the truths of basic arithmetic – i.e. the truths expressible in terms of successor, addition, multiplication, and the apparatus of first-order logic. Ideally, we'd like to construct a consistent theory T whose language includes L_A and which proves all the truths of L_A (and no falsehoods). So we'd like T to be negation complete, at least for sentences of L_A. But, given some entirely natural assumptions, there can't be such a negation-complete theory.

The first natural assumption is that T should be set up so that it is effectively decidable whether a putative T-proof really is a well-constructed derivation from T's axioms. So, in short, we want T to be a properly axiomatized theory. Indeed, we surely want more: we want it to be decidable what counts as a T-proof without needing open-ended search procedures (if would be a very odd kind of theory where, e.g., checking whether some wff is an axiom takes an unbounded search).