The transitive multiverse

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins’ multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, $\{(M,E)\}$ would be an entire multiverse. That’s quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit… off putting.

There is also Väänänen’s multiverse, which is more abstractly defined, and I cannot for the life of me recall its definition and its details.

Some time ago Ur Ya’ar gave a seminar talk about Hamkins’ multiverse in the logic seminar in Jerusalem. It was interesting, and afterwards Yair Hayut and myself talked with Ur about these multiverses. One idea that came up, and I don’t think that I ever ran into it, is sort of a combination between the generic multiverse and Hamkins’ multiverse. Consider the following axiom “Every real is an element of a transitive model”. Now look at $\cal M$, the set of all the countable transitive models, we get the following axioms are satisfied by $\cal M$:

  1. If $M\in\cal M$, then every generic extension and every ground of $M$ is also in $\cal M$.
  2. If $M\in\cal M$, then every inner model of $M$ is also in $\cal M$.
  3. If $M\in\cal M$, then there is some $N\in\cal M$ such that $M\in N$ and $N\models M\text{ is countable}$.
  4. For all $M,N\in\cal M$, $L^M$ and $L^N$ are comparable.

So what do we have here? We have a multiverse of sets, it is closed under generic extensions and grounds, and it is even closed under definable inner models. It also has the property that we can always find bigger models that think a given model is countable.

Now, I have no idea what useful things can come out of this multiverse. And I would imagine that one should first refine this notion a bit more before it becomes actually useful for something. But nonetheless, it seems like an interesting interpretation of the whole notion of multiverse.

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