23 - Interlude: About the First Theorem

Summary

We have achieved our initial goal, namely to prove Gödel's First Incompleteness Theorem, and to do so in Gödelian style. Other proofs will come later. But it will do no harm to pause to survey what we've established so far 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. Finally in this interlude, we will outline the next block of chapters which explore further around and about the First Theorem.

What we have proved

We begin, then, 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, meaning the truths expressible in terms of successor, addition, multiplication, together with the apparatus of first-order logic (so these are the truths that can be rendered in L_A). Ideally, we would like to construct a formally defined consistent theory T whose language includes L_A and which proves all the truths of L_A (and which entails no falsehoods). So we would like T to be negation-complete, at least for sentences of L_A. But the First Theorem tells us that, given some entirely natural assumptions, there just can't be such a negation-complete theory.

The first required 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.