Naive set theory entails contradictions. Really bad news. Or so most of us think. But what if we are prepared to be more tolerant of contradictions, e.g. by adopting a dialethic and paraconsistent logic, which allows there to be contradictions which are true (as well as false) and where contradictions don’t entail everything? Could we rescue the naive conception of set, accommodate e.g. the idea of a Russell set, by departing from classical logic in this way? A desperate measure, most of us will think. Even if willing, once upon a time, to pause to be amused by varieties of dialethic logic, at this late stage in the game, I don’t have much patience left for the idea of going naive about sets by going far-too-clever-by-half about logic. But Luca is evidently a lot more patient that I am! He devotes Chapter 4 of his book to investigating various suggestions about how to save naive set theory by revising our logic. How does the story go?
Luca very helpfully divides his discussion into three main parts, corresponding to three dialethic strategies.
The first he labels the The Material Strategy — we adopt a non-classical logic which keeps the material conditional, so that $latex P \to Q$ is still simply equivalent to $latex \neg P \lor Q$, while we reject the classically valid disjunctive syllogism, and hence material modus ponens.
Graham Priest initially thought that a ‘simple and natural choice’ here is his LP, the Logic of Paradox. But LP doesn’t validate the transitivity of the material conditional, and this hobbles the proof of various elementary theorems of set theory — even the usual proof of Cantor’s Theorem fails. And on the usual definition of set identity in terms of coextensionality, Leibniz’s Law fails too.
Can we tinker with LP to avoid these troubles? Luca mentions a few options: all of them have equally unattractive features — giving us a set theory that is too weak to be useful. So, Luca’s verdict seems right: the prospects of saving naive set theory as a formal theory by developing the material strategy in anything like the way originally suggested by Priest are dim indeed.
What about the second strategy The Relevant Strategy, where we adopt a relevant logic that does validate modus ponens? Luca dives into an extended discussion of so-called depth-relevant logics, also advocated by Priest. We won’t follow the ins and out of the arguments here. But it is again hard not to agree with Luca’s eventual verdict that (i) “Priest does not seem to have offered a good argument for focusing on the logic he is considering,” and further (ii) the logic is in any case still too weak for us to carry out standard arguments in set theory. At which point, most of us would get off this particular bus! But Luca takes us on another couple of stops, considering some proposals from Zach Weber for strengthening the depth-relevant logic. The argument become more involved, but again Luca arrives at a strongly negative verdict (enthusiasts can follow up the details): Weber’s logic remains poorly motivated — I would say it smacks of ad-hoc-ery — and Weber’s resulting theory lacks a genuine principle of extensionality so in the end can’t be regarded as a set theory anyway.
That leaves us with a third strategy also to be found in Priest, The Model-Theoretic Strategy. An intuitionist can allow that classical logic is just fine when we are reasoning about decidable matters. Similarly, the idea now is, that a dialetheist can allow that classical logic is just fine when we are reasoning over consistent domains. So the dialetheist can continue to reason classically about a consistent core subdomain of sets — e.g. the cumulative hierarchy — while asserting that this is just part of the full universe of naive set theory, about which we must reason dialethically.
There is, you might well think, an immediate problem about this position. If you accept the cumulative hierarchy as a consistent core universe of sets about which we can argue classically, and which gives us all the sets we can want for mathematical purposes (ok, push the hierarchy up high enough to keep the category theorists happy!), then just what are we getting by adding by more putative sets, the inconsistent ones? It’s not like adding points at infinity to the Euclidean plane, for example, to give us an interestingly enriched mathematical structure (results about which can be reflected down into interesting new results in the original domain). And I’m ‘naturalist’ enough in Maddy’s sense to ask: if the putative enriched structure of naive set theory isn’t mathematically interesting, why bother?
But I’ve jumped the gun: what does this supposed enriched structure look like? With some trickery which Luca describes, we can explain how to extend a model of ZF to become a model of a naive set theory with LP as its logic (so the formal theory might be inadequate to work inside, as argued before, but still have a rich model). But why suppose that this particular fancied-up model captures the structure of the full supposed universe of a naive set theory? As Luca points out, we have no good reason to suppose that. But then “the paraconsistent set theorist needs, after all, to say more about what the universe of sets looks like. It is not enough to simply suppose that it contains some paradigmatic inconsistent sets and has the cumulative hierarchy as an inner model.” The story is radically incomplete.
Summing up, the idea of rescuing naive set theory by going dialethic, using any of the three strategies, looks to be degenerating research programme. Probably many of us would have predicted that outcome! — but we can thank Luca for doing the hard work of confirming this in some detail.
And that takes us to the end of Chapter 4, and a bit more than half way through the book. And for most readers, the three remaining chapters promise to be considerably more interesting than the one we’ve just been looking. Next up, for example, is a discussion of the commonly-encountered idea of ‘Limitation of Size’.