Tonight I was fondly recalling Michael Hand and Jonathan Kvanvig's old paper on Tennant's solution to Fitch's Paradox (it's a beautiful read) and a weird thought occured to me.
Hand and Kvanvig argue that Tennant's solution would be analogous to a set theorist responding to Russell's Paradox by proposing naive set theory with Frege's comprehension axiom restricted to instances that don't allow one to prove absurdity from that instance and the other axioms (call this theory N'). For Hand and Kvanvig this is a reductio of Tennant, and in my response* I argued that Tennant's solution was not actually analogous to N'.
If I remember right, Hand and Kvanvig argue that N' is bad because it doesn't illuminate the nature of sets in the way we properly expect of solutions to paradoxes. But they don't go into the logical properties of N' at all, and tonight I'm thinking that this is actually an important question in its own right. Let's just consider consistency, completeness, and axiomatizability.
Is N' consistent? I can't prove this (I'm in that penumbral region between not being a logician and being a terrible one), but it looks consistent. Could some finite (because compactness) set of instances of comprehension that on their own don't entail absurdity jointly entail absurdity? If not, then N' is consistent. If the problem does occur, then one could propose N'', which is naive set theory with every instance of comprehension that does not entail absurdity when paired with any finite set of instances of comprehension that do not entail absurdity on their own. In what follows I'll assume that N' is consistent, but if not the following questions could be raised about N''.
Is N' incomplete? Again I don't know how one might prove that. You couldn't do kind of proof I'm familiar with unless you knew that the set of sentences in N' were recursively enumerable, or equivalently by Craig's Theorem, axiomatizable.
There's an intuitive sense in which N' is complete. Arguably, the whole purpose of non-naive set theory has been to axiomatize a system that gets you as much of comprehension as you can get without generating contradictions. If that's correct, then (assuming it's consistent) a theory that generates the truths of N' is the theory that people have been aiming at.
But then this raises the question of whether the goal is achievable. That is, assuming N' is complete, is it axiomatizable? It has the feeling of not being so. Consider the analogous (albeit inconsistent) theory where you take all of the sentences that prove absurdity out of the language of first order logic, so what you are left with are the logical truths and logical contingencies. This theory is provably non-enumerable/unaxiomatizable. If it were enumerable, then the set of inconsistent sentences would be decidable (since that set is already enumerable), but they provably aren't.
Again though, the analogous argument would only work if we knew computability type facts about the set of sentences that entail absurdity in naive set theory.
In any case, if the correct set theory coincides in the set of truths with N', and N' is not axiomatizable, then the correct set theory would not be axiomatizable. If, big if, that were the case, then Russell's paradox showed us something different from what we normally take to be the case.**
[Notes:
*I wish I'd understood Graham Priest's work as I do now when I wrote the section showing the similarities between Fitch's proof and the paradox of the stone, an analogy on which my defense of Tennant stands. I keep meaning to go back and respond to Timothy Williamson's argument against Tennant, which is interesting in its own right. I even had half a paper written at one point, but over the last decade I kept getting sidetracked by other projects.
**As noted above, I'm not a logician. When I get a weird idea like this, it's only around half the time there's something interesting in it that hasn't been published already. But, given Tennant's work, any corrections and/or pointers to relevant articles would be really helpful for thinking more deeply about Fitch's paradox in light of Hand and Kvanvig's analogy.]
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