14 - Interlude: A very little about Principia

Summary

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'll review the stages that remain.

But first, let's relax for a moment after our labours, and take a very short look at some of the scene-setting background. We'll say more about the historical context in a later Interlude (Chapter 28). But for now, we ought at least to 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

As we noted in Section 10.8, Frege aimed in Grundgesetze der Arithmetik to reconstruct arithmetic on a secure footing by deducing it from logic plus definitions. But – in its original form – his overall logicist project flounders on Frege's fifth Basic Law, which leads to contradiction. And the fatal flaw that Russell exposed in Frege's system was not the only paradox to bedevil early treatments of the theory of classes.