The transitive multiverse

Jul 22 2017, 22:18

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.