I present a construction of a model of an expansion of set theory. It embodies a conception of a multiverse of universes of set theory, with an untyped notion of truth-relative-to-a-universe. The construction is partly motivated by philosophical concerns, which I am in the process of pondering. As this is work in progress, I'm keen to hear your feedback. 
 
As a foundation for mathematics, set theory is a formal system for constructing and reasoning about an extremely wide range of abstract objects. However, it follows from Gödel's second incompleteness theorem that no such foundational system grants us the power to construct a model of that system itself. Given this situation, what is the significance of having proofs from the axioms of a set theory such as ZFC? Here are three potential answers to this question:
 
1. The conclusions of proofs from ZFC are true about sets.
2. The conclusions of proofs from ZFC are true in every structure satisfying ZFC. 
3. For a proof p from ZFC to gain significance it needs to be interpreted in a particular mathematical structure, e.g. as establishing a fact about the real numbers. 
 
A philosophical approach to these matters that I am probing in this seminar is to adopt a point of view influenced by Carnap, with a distinction between internal and external notions: A problem with (1) is that its notion of truth is external and therefore dubious. (2) looks promising in the sense that the proofs gain significance internally to each model of ZFC, but the problem is that ZFC does not prove that there are any such models. In (3), ZFC takes the rôle of an umbrella, under which we can fit many distinct mathematical structures, and a proof is only significant as interpreted internally to such a structure. On this view it might be tempting to dismiss the question of the significance of proofs from ZFC, as such, as an external and therefore senseless question. However, I propose that a way forward is to internalize (1).
 
We may internalize the notion of truth by adding a predicate of truth with natural axioms to ZFC. A robust manner of doing so is to add compositional axioms of truth, resulting in the theory CT(ZFC) . Conveniently, CT(ZFC) proves the statement (known as "global reflection over ZFC") that the consequence of each proof from ZFC is true, thus establishing (1) as a formally proved statement. In light of this result, let us denote CT(ZFC) as GR. GR shows not only that there are models of ZFC, but that there are some particularly interesting ones, thus solving the problem with (2). However, this move begs the question about the significance of proofs from GR. Iterating this process we can add the axiom of global reflection over GR to GR, obtaining the system GR^2. Next we obtain the system GR^3, and so on towards infinity. We can take the limit at infinity, obtaining the theory GR^omega. It turns out that the theory GR^(omega+1) is inconsistent, thus "stopping the regress" at this point.
 
I will explain my construction of a multiverse of universes of set theory in GR^omega and a revision semantics for truth-relative-to-a-universe in this multiverse. Time permitting, I'd be happy to discuss philosophical aspects of this "reflective multiverse" conception of set theory.