A natural model of the multiverse axioms

This is a talk at the MIT Logic Seminar, April 8, 2010.
Slides

In the past decades, set theorists have studied a multitude of set theoretic universes constructed as forcing extensions or sub-models and often incorporating strong axioms of infinity. Cohen’s forcing was generalized to build extensions according to nearly any desired recipe, while Gödel’s construction of L was generalized to obtain increasingly sophisticated sub-models. Against this background, it can be argued that set theory moved away from the study of the set theoretic universe to that of the set theoretic Multiverse. Philosophically, a multiverse encompasses a rich collection of possible ZFC-universes. It should include universes satisfying or failing to satisfy the Continuum Hypothesis, Martin’s Axiom, $\diamondsuit$-principle, etc. Some universes in the multiverse should be countable from the perspective of other universes, some ill-founded, and some ultrapowers. The multiverses admit a purely mathematical, non-philosophical treatment as mathematical objects in ZFC. We define that a multiverse is simply a nonempty set or class of models of ZFC. The Multiverse Axioms for such collections are intended to capture the richness and interconnectedness of the ZFC worlds. They include:

In this talk, I will introduce the multiverse axioms and show that if ${\rm ZFC}$ is consistent, then the collection of all countable computably saturated models of ZFC is a natural model of the multiverse axioms. A model is computably saturated if it already realizes all its finitely realizable computable types. This is joint work with Joel David Hamkins.