Sometimes our world is said to be a "Big Conjunctive Contingent Fact" or that other possible worlds are "recombinations" of available propositions for some actual world. So model-theoretically, a universe can be taken for a set of propositions. So if models and model-theoretic universes can nontrivially embed into each other and/or themselves in various ways, can physical universes and subuniverses do the same?

Motivation: the critical points of various nontrivial universal/subuniversal embeddings (in a priori logical space) have a property called strong compactness which involves limits on the number of logical operators and quantifiers sustained by sentences of the models/universes. If a physical world at some time t was strongly compact modulo some limits k, and then proceeded to embed itself into a subworld at later(t), its compactness signature would upwardly change, too, then, or the size of the Big Conjunctive Contingent Fact constitutive of that world would grow.

Quine thought that the mathematics provided by the axiom of constructibility should be good enough for us to apply to the physical sciences. So a Quine-Smolin dynamic might be framed in terms of relative constructibility, where constructible subworlds embed into themselves, generating new sets of order-indiscernibles along the way, but otherwise we would not there worry about the entire universe embedding itself into itself or any subuniverses. On the other hand, I don't know if the initial critical point of j: L → L is strongly compact (it seems like it wouldn't be, since it wouldn't be measurable, but if it was strongly compact it would be measurable?), although perhaps it could be weakly compact instead (and then our concern would be with a Big Conjunctive Contingent Fact evolving in size to conform to the upward parameters of weak compactness).

Objection?: however, I was wondering how to square the multiverse standpoint with the idea of evolving worlds, and I wondered whether elements assigned to some universe could, as time goes on, be reassigned to more complex universes. Again, we would not seem to need to detain ourselves with worrying about the dynamics of the entire multiverse (since there isn't a singular "entire multiverse" anyway, at least not on every logical level), or then no elements are always elements of some universe and the multiverse overall. We could still overlap the embedding flux and the assignment flux, maybe (the elements are switched from a world lacking some specific embedding, to a world having that specification, and so on).

Would a hypothesis of an evolving individual universe (where the universe embeds into subuniverses, if not itself) be more or less absurd/logically (im)possible than a hypothesis of sets of masses from various universes periodically teleporting en masse into other universes? (C.f. the question of causal closure as a characteristic parameter of singled-out worlds.)

  • A model universe is a representation of a physical universe. Asking whether or not models can embed themselves in other models is a completely different thing than asking whether or not physical universes can do so. I can cut a road map with scissors and repair it with tape but I can't do that to the landscape. Commented Jul 4, 2023 at 20:05
  • @nielsnielsen granted, but oddly, the Cantor's Attic article on strong compactness includes a section on topology/subspaces, so I wonder... Moreover, there's got to be something behind Hamkins/et. al.'s talk of machines with buttons and switches and ratchets, and set-theoretic geology, that resonates with things like set-theoretic potentialism. If V even without being a model can self-embed nontrivially (or embed into other classes nontrivially), then I wonder if the dynamic metaphors for V can cross over to physical space better (Corazza appears to think so, FWIW). Commented Jul 4, 2023 at 20:59


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