36 - Looking back

Summary

Let's finish by taking stock one last time. At the end of the last Interlude, we gave a road-map for the final part of the book. So we won't repeat the gist of that detailed local guide to recent chapters; instead, we'll stand further back and give a global overview. And let's concentrate on the relationship between our various proofs of incompleteness. Think of the book, then, as falling into three main parts:

(a) The first part (Chapters 1 to 7), after explaining various key concepts, proves two surprisingly easy incompleteness theorems. Theorem 5.7 tells us that if T is a sound axiomatized theory whose language is sufficiently expressive, then T can't be negation complete. And Theorem 6.2 tells us that we can weaken the soundness condition and require only consistency if we strengthen the other condition (from one about what T can express to one about what it can prove): if T is a consistent axiomatized theory which is sufficiently strong, then T again can't be negation complete.

Here the ideas of being sufficiently expressive/sufficiently strong are defined in terms of expressing/capturing enough effectively decidable numerical properties or relations. So the arguments for our two initial incompleteness theorems depend on a number of natural assumptions about the intuitive idea of effective decidability. And the interest of those theorems depends on the assumption that being sufficiently expressive/sufficiently strong is a plausible desideratum on formalized arithmetics.