Set-theoretical multiverse and foundations I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, but I was fascinated by the possible philosophical perspective of being compelled (by mathematical practice of forcing in set theory) to consider a whole multiverse of sets as a natural "landscape" for set theory, without committing to any specific choice.  
[There's a kind of abstract in the introduction of a n-category café blog post (which I haven't completely read yet) by David Corfield]
In the article it is also stated that it's possible to mimick the study of the "full multiverse" within ZFC. This is actually done in A natural model of the multiverse axioms: "we shall internalize the study of multiverses to set theory by treating them as mathematical objects within ZFC [...]" 
Personally, I have more affinity for the formalistic viewpoint than for some version of platonism (multiversed or not). So, my first question (somehow dually to this one) is:

Is it conceivable that the "set theoretic multiverse principles" (which at the moment are, properly, ZFC sentences - see Hamkins and Gitman-Hamkins) could fit into a formal "multiverse theory" which is carried out in its own, i.e. not within a metatheory like ZFC, hence capturing the full-blown multiverse? Could such a theory be taken as the foundation (at least in some ZFC-flavoured sense) of mathematics?

(I will probably open one or more followup "philosophical" questions about multiverses, when I'll clarify to myself what to ask)
 A: (It happens that I will be giving a talk this week on this topic at the Exploring the Frontiers of Incompleteness series at Harvard, which is focussing on the question of determinism in set theory. The materials for all the talks are posted on the EFI web page, which also includes videos and discussion forums.)
Set theory currently provides at least a robust informal treatment of the multiverse. What I mean by this is that it is part of standard set theoretic practice and understanding to point out when a set-theoretic assertion $\varphi$ can have different truth values in different set-theoretic universes. That is, set theorists are already paying attention to the multiverse, when they discuss such issues as independence, forcing absoluteness, indestructibility and so on, which all concern the issue of how set-theoretic assertions can vary in the mutliverse. 
Another more formal treatment of the multiverse idea is provided by what I have described as the toy model formalization, where one considers a multiverse from the perspective of a much larger set-theoretic background. For example, in ZFC one may consider the multiverse of all countable models of set theory, or just of a portion of them. This is the approach that Gitman and I took in our consistency proof that you mentioned, A natural model of the multiverse axioms. One can also view the typical countable-transitive-model approach to forcing as an instance of the toy model formalization. 
Perhaps the toy model formalization has some affinity with the idea of the axiom of universes in category theory, and such a formalization would lead to a tower of multiverse concepts, one existing at the level of each Grothendieck universe. So it seems clear that one may set up a formal theory of multiverses using ZFC as a background theory, supplemented with various universe axioms, and leading to a tower of multiverses.
But my opinion is that the toy model method of formalization is ultimately unsatisfying, since it leads one to adopt axioms and principles about the toy models, rather than about the background set theory, which is the intended goal. 
Although many of the questions that we have about the full multiverse happen to be first-order expressible in the language of set theory, nevertheless many of the questions that we are interested in happen to be formalizable in first-order ZFC. For example, many of the concepts of set-theoretic geology are formalizable in first-order ZFC, such as the open question whether every two ground models of the universe have a deeper ground inside them. Similarly, the questions surrounding the modal logic of forcing are also first-order expressible. This is a happy phenomenon, when we are able to use our current formalization to express and understand the multiverse issues. But the troubling matter underlying your question is that some of our questions are indeed pushing up against our formalism, which may be inadequate to the task. 
