The title of Gödel's great paper is ‘On formally undecidable propositions of Principia Mathematica and related systems I’. And as we noted in Section 23.4, his First Incompleteness Theorem does indeed undermine Principia's logicist ambitions. But logicism wasn't really Gödel's main target. For, by 1931, much of the steam had already gone out of the logicist project. Instead, the dominant project for showing that classical infinitary mathematics is in good order was Hilbert's Programme, which we mentioned at the outset (Section 1.6). This provided the real impetus for Gödel's early work; it is time we filled out more of the story.

However, this book certainly isn't the place for a detailed treatment of the changing ideas of Hilbert and his followers as their ideas developed pre- and post-Gödel; nor is it the place for an extended discussion of the later fate of Hilbertian ideas.1 So our necessarily brief remarks will do no more than sketch the logical geography of some broadly Hilbertian territory: those with more of a bent for the history of logic can be left to fight over the question of Hilbert's precise path through the landscape.

Another, quite different, topic which we will take up in this Interlude is the vexed one of the impact of the incompleteness theorems, and in particular the Second Theorem, on the issue of mechanism: do Gödelian results show that minds cannot be machines?