## Luca Incurvati’s Conceptions of Set, 6

In §3.3 of Conceptions of Set, Luca discusses what he calls the ‘no semantics’ objection to the iterative conception. He sums up the supposed objection like this:
Consider the case of iterative set theory, which for present purposes will be our base theory Z+. Since the set-theoretic quantifier is standardly taken as ranging over all sets, it seems that one of the interpretations quantified over in the definition of logical validity for L [the standard first-order language of set theory] – the intended interpretation – will have the set of all sets as its domain. But there can be no set of all sets in Z+, on pain of contradiction. Hence, the objection goes, if we take all sets to be those in the hierarchy, we cannot give the usual model-theoretic definition of logical validity.
Or rather, that is how the objection starts. Of course, the further thought that is supposed to give the consideration bite is that, if we can’t apply the usual model-theoretic definition of logical validity, then we are bereft of a story to tell about why we can rely on the inferences we make in our set theory when we quantify over all sets.
As Luca immediately remarks, this challenge is not especially aimed at the iterative conception: any conception of the universe of sets that rules out there being a set of all sets will be open to the same prima facie objection. It looks too good to be true!
Graham Priest is mentioned as a recent proponent of this objection. But as Luca point out, Kreisel over fifty years ago both mentions the issue raised in the quote and has a response to what I called the further thought which is supposed to make the issue a problem. For Kreisel’s ‘squeezing argument’ is designed precisely to show that we have a perfectly good warrant for using standard first-order logic as truth-preserving over all structures, not just the ones that can be formally regimented in the usual model-theoretic way. I’ve defended Kreisel’s argument, properly interpreted, e.g. here: so I’m. . .

News source: Logic Matters