This is a contributed talk at the RIMS Set Theory Workshop 2019. It is about joint work with Victoria Gitman, Michał Tomasz Godziszewsky, and Toby Meadows.
The multiversist position in the philosophy of set theory holds that rather than there being an absolute notion of set, there are instead many universes of set theory, each of equal ontological status. Hamkins proposed a series of Multiverse Axioms to capture his position on the structure of the set theoretic multiverse, with his Well-Foundedness Mirage axiom being the most provocative. Gitman and Hamkins showed that the collection of countable, recursively saturated models of set theory form a multiverse satisfying Hamkins’s Multiverse Axioms. And it is easy to check that if a multiverse of models of set theory satisfies the Well-Foundedness Mirage axiom then every world in that multiverse must be recursively saturated.
Gitman, Godziszewski, Meadows, and I investigated whether this forced recursive saturation could be avoided by weakening the Well-Foundedness Mirage axiom. We considered two weakenings, and showed that neither of them forces all worlds in a multiverse to be recursively saturated. In this talk I will discuss the construction of our two multiverses.