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In, "Multiversism and Concepts of Set: How much relativism is acceptable?" Neil Barton distinguishes between an ontological interpretation of set-theoretic multiverses as referents and an algebraic interpretation that is very much like, or is just a version of, if-thenism, which countenances mathematical abstractions as true (or at least nontrivially justifiable) conditional sentences:

Under the algebraic interpretation, we do not take set theory as the sort of enterprise that is concerned with referring to objects. Rather it is seen as providing an intuitive framework that underlies an algebraic method of thinking. This then allows us to understand what will be possible on a given structure with certain properties. Thus the problem of reference dissolves: we are not even making the appropriate kinds of claims to be assessed for reference.

What struck me about the gist of the essay as a whole was how much it reminded me of the actualist-possibilist debate, esp. the question of the existential Barcan formula:

  1. ◊∃νφ → ∃ν◊φ?

Maybe this issue is addressed somewhere in one of the essays about set-theoretic potentialism that I've sifted through (e.g. in denying either (1) or its converse), though my memory isn't telling me so right now. So my question is whether the algebraic multiverse is ontologically noncommittal in the sense of being, for set-theoretic possible-worlds talk, what actualism is for basic possible-worlds talk? I.e., talk of set-theoretic multiverses can use "(relatively) maximal consistent sets of propositions" as individual universes formulated over only one "actual" universe (a core model? set-theoretic geological mantle? I still don't understand those notions very well, if at all), whereas the ontological multiverse involves a set-theoretic Barcan formula that describes its intended referential value.

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So complex question (and I am very rusty), but certainly there is a very real branch in the Hamkins style Set theoretic Multiverse that wants to treat the plurality of set theoretic worlds as a kind of Modality, in just the same sense that we use in the metaphysics of possibility and necessity, and hence something about which some kind of Modal Logic might exist.

In his A Simple Maximality Principle, Hamkins took up this idea by talking about statements that were Necessary under forcing extension, and exploring the consequences of certain modally natural principles on the resulting logics and the strengths of the set theories they chart. So there's something to be said for the parallel between a plurality of set theoretic universes and the idea of ontologically distinct mereological wholes (to borrow the Lewis coinage).

However, there is one way in which they might come apart. I think for Hamkins, as Neil discusses in his paper, there's probably no privileged sense of the "Actual" set theoretic universe, even situated within a multiverse, because of the large overlap potential of forcing extensions and the potential for reversibility. However, some forcing extensions are a bit "stickier" and harder to undo than others, such that some of the multiverses introduce consequences that weigh in on the modality in question, making it difficult to pin down exactly what the gap between the modal logic and the set theory should amount to and no obvious way to answer it.

That's okay, because set theoretic "possibility" doesn't necessarily need to overlap with metaphysical possibility - the full set theoretic multiverse might actually exist metaphysically in its entirity, in the ontological sense, as well as each metaphysically possible world containing its own Set theoretic multiverse.

A really radical Formalist, Actualist view of the set theoretic modality might flip this completely on its head, though, and say "Yes, this modalizing view is useful, because the modality at work here is exactly the kind of capability model that we reference elsewhere in our semantic theorising rather than an expansion of ontologies - multiverse talk is a really conceptually useful extrapolation/development of the technology of Sets, and as long as we can derive interesting and true maths from it then it's fine to be a discursive tool independent of the metaphysics".

I don't think this is what Barton's Algebraic multiverse theorist is quite saying (though Hilbert is definitely present in all of this). His view is more Structuralist - the multiverse theory of higher set theory outlines the skeleton of an interesting mathematical complex, a kind of local geometry relative to the set theories we find ourselves in about which principles and axioms can be stated. Those complexes might also be realized as a matter of the larger multiverse-level scale, and if so then we might also hold the ontological interpretation to have been satisfied as well, but that doesn't mean we have to wait to that point to use the forcing modal view as a valuable tool within and between set theories.

Again, not quite sure if I've got the point exactly, very much out of step with the literature, but think there's a lot of unexplored potential in there!

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    You do understand the material. I think I forgot to check if Barton ended up being more favorable towards a sort of structuralist realism in between the Hamkins multiverse and a "merely" algebraic one, and I think he means this as an option when he says how the multiverse standpoint can be ontologically sustained assuming at least a few absolutes. He leans towards the "merely algebraic" picture, but not necessarily as a final say but just as a reasonable dialectical move (his essays about these concepts are ongoing and intriguing, e.g. his new one about mathematical Gettier cases). Jun 7, 2023 at 23:13

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