8 - Interlude: Taking stock

Summary

Comparing incompleteness arguments

Our informal incompletability results, Theorems 6.3 and 7.2, aren't the same as Gödel's own theorems. But they are close cousins, and they seem quite terrific results to arrive at so very quickly.

Or are they? Everything depends, for a start, on whether the ideas of a ‘sufficiently expressive’ arithmetic language and a ‘sufficiently strong’ theory of arithmetic are in good order. Still, as we've already briefly indicated in Section 3.1, there are a number of standard, well-understood, ways of formally refining the intuitive notions of effective computability and effective decidability, ways that turn out to specify the same entirely definite and well-defined class of numerical functions and properties. Hence the ideas of a ‘sufficiently expressive’ language (which expresses all computable one-place functions) and a ‘sufficiently strong’ theory (which captures all decidable properties of numbers) can in fact also be made perfectly determinate.

But, by itself, that claim doesn't take us very far. For it leaves wide open the possibility that a language expressing all computable functions or a theory that captures all decidable properties has to be very rich indeed. However, we announced right back in Section 1.2 that Gödel's own arguments rule out complete theories even of the truths of basic arithmetic. Hence, if our easy Theorems are to have the full reach of Gödel's work, we'll really have to show (for starters) that the language of basic arithmetic is already sufficiently expressive, and that a theory built in that language can be sufficiently strong.