A Question Regarding an Accessibility Relation on the Class of All Models of ZFC

Question

Consider the class of all models of ZFC and consider the following accessibility relation on the class--that each model in the class can know what are the cardinals and ordinals of every other member of the class, what are the sets of ordinals of every other member of the class, and what is the satisfaction relation of every other member of the class (it is a given that this accessibility relation is reflexive and it is also a given that beings within each model can communicate with beings in every other model in the class). Certainly the class of all models of ZFC with this accessibility relation forms a type of 'multiverse' (by definition of the accessibility relation, this class with this type of accessibility relation will form the semantics of the modal logic S5 and should satisfy some sort of maximal principle). What would this maximal principle be and what happens to forcing within this 'multiverse'--in particular, what happens to cardinal collapse within this 'multiverse' (i.e. can cardinal collapse occur within this type of 'multiverse')?

Answer 1

Thomas, I'm glad to hear that you're reading my papers. Let me try to answer your question, which I've just noticed.

The first thing to say is that you don't seem to be using the accessibility concept as it is normally understood. A Kripke model consists of a collection of possible worlds, each with its own truths in the underlying (nonmodal) language, and a relation between the worlds called the accessibility relation. Thus, we say that one world $u$ in such a Kripke model accesses another world $v$, if $u$ is related to $v$ under this accessibility relation. The accessibility relation is then used to define the truth condition for modal assertions at each world. Namely, the assertion $\diamond\varphi$ is true at a world $u$, just in case $\varphi$ is true at some world $v$ that $u$ can access. So the worlds of the Kripke model become a directed graph under the accessibility relation, and the nature of this directed graph often determines the modal theory. For example, when the digraph is a partial order, then you'll get S4 being true at every world, and if it is reflexive, transitive and symmetric (i.e. an equivalence relation), then you get S5 being true at every world.

But the accessibility relation you mention in your question is not like this, since it is not a relation between worlds. Rather, you seem to have in mind some kind of relation that allows one model of set theory to see inside another model and inspect the nature of its ordinals, cardinals and sets of ordinals. But if it is allowed to undertake constructions with the knowledge it gains from such kind of inspections, and form new sets using those predicates, then this will definitely violate ZFC. For example, no model will be able to satisfy the power set axiom, since there will always be some other world with a new subset added by forcing, and with the kind of access you seem to mention, the new set would have to have been there already, violating power set.

So your construction does not seem to be set-up for the maximality principle, without further explanation about what you meant.