Luca Incurvati’s Conceptions of Set, 10

I’m picking up Luca’s book again, at Chapter 5. In the previous chapter, the question was: can we save the naive conception of set from ruin by tinkering with our logic? Short answer: no, not in a well-motivated way that will leave us with a set theory worth having. In this chapter, the question is: can we save the essence of the naive conception while retaining classical logic by minimally restricting naive comprehension?

Quine wrote:

Only because of Russell’s paradox and the like do we not adhere to the naive and unrestricted comprehension schema […] Having to cut back because of the paradoxes, we are well advised to mutilate no more than what may fairly be seen as responsible for the paradoxes.

Which suggests a simple-minded approach. Take one step back from disaster, and just accept all the instances of comprehension that do not generate paradox.

But what would that mean? First option: we should severally accept each instance of comprehension that does not entail contradiction. But it is easy to see that this is won’t work, because instances of comprehension which — taken separately — are consistent can together entail contradiction. Luca gives nice examples.

Second option, and surely more in keeping with Quine’s intention: we should accept those instances of comprehension which taken together do not entail contradiction. But this idea too doesn’t work.

As Luca points out, the proposal now is reminiscent of another paradox-avoiding proposal: be almost naive about truth by accepting just the maximal consistent set of all the instances of the T-schema. But McGee has a nice argument showing that that idea won’t wash; and Luca with Julien Murzi has generalized McGee’s argument so it applies here.

As a warm up, we can show that a maximally consistent set of instances of naive comprehension that e.g. includes the claim that there is an empty set is negation complete and hence (assuming the resulting set theory interprets Robinson arithmetic, a very weak demand!) is not recursively axiomatizable. Now, Luca says that such a set of instances can then hardly count as a set theory; but I’m not quite sure how much that would worry Quine. OK, any nicely axiomatized subset of that maximally consistent won’t be the full story about sets; but if we regiment enough of those consistent instances of comprehension into a theory rich enough for the working mathematician’s ordinary set-theoretic purposes (extending the theory if and when needed), why worry? We have a working theory (in one familiar sense), and a supposed story about why it is nice, i.e. it is part of the maximal naive set theory (in another familiar sense).

But let’s not pause over the hors d’oeuvre: the main course is the demonstration that (as in the case with instances of the T-schema) there after all there isn’t a unique maximal consistent set of all the instances of naive comprehension. Well, you might think that that too wouldn’t be too disastrous if the possible divergences between consistent stories were remote from ordinary business. But no such luck — uniqueness fails as badly as possible:

For any consistent sentence σ, there is a maximally consistent set of instances of naive comprehension implying σ, and another one implying ¬σ.

So the proposal to look for a maximal consistent set of instances of naive comprehension fixes no determinate theory at all (axiomatizable or otherwise). Hence the simple-minded reading of Quine’s one-step-back-from-disaster is indeed simply hopeless.

Two pernickety quibbles. The McGee 1992 paper in the biblio is the wrong one, it should his ‘Maximal Consistent Sets of Instances of Tarski’s Schema (T)’ in JPL. And the beginning of the Appendix on the Incurvati/Murzi upgrade of McGee’s theorem isn’t as clear as it ideally could be: e.g. a reader will pause to wonder what $latex \mathsf{S}^{\prime}$ is. But the argument of §5.1 indeed seems conclusive. The rest of this chapter is then about  ‘limitation of size’ principles. So let’s pause here and consider them in the next post.