In the last Interlude, we gave a five-stage map of our route to Gödel's First Incompleteness Theorem. The first two stages we mentioned are now behind us. They involved (1) introducing the standard theories Q and PA, then (2) defining the p.r. functions and – the hard bit! – proving Q's p.r. adequacy. In order to do the hard bit, we have already used one elegant idea from Gödel's epoch-making 1931 paper, namely the β-function trick. But most of his proof is still ahead of us: at the end of this Interlude, we will review the stages that remain.

But first, let's relax for a moment after all our labours, and pause to take a very short look at some of the scene-setting background. We will say more about the historical context in a later Interlude (Chapter 37). But for now, we'll say enough to explain the title of Gödel's great paper: ‘On formally undecidable propositions of Principia Mathematica and related systems I’.

Principia's logicism

Frege aimed in his Grundgesetze der Arithmetik to reconstruct arithmetic (and some analysis too) on a secure footing by deducing it from logic plus definitions. But as we noted in Section 13.4, Frege's overall logicist project – in its original form – founders on his disastrous fifth Basic Law. And the fatal contradiction that Russell exposed in Frege's system was not the only paradox to bedevil early treatments of the theory of classes.