Set-theoretic geology: Excavating a local neighborhood of the multiverse
CUNY New York City College of Technology
This talk will give a brief overview of set-theoretic geology, the study of the collection of grounds of $V$. Forcing is naturally viewed as a method for passing from a model $V$ of set theory (the ground model) to an outer model $V[G]$ (the forcing extension). A change in perspective, however, allows us to use forcing to look inward: from a model $V$, we define an inner model $W$ of $V$ to be a ground of $V$ if $W$ is a transitive proper class satisfying ZFC and $V$ can be obtained by forcing over $W$, that is, if $V = W[G]$ for a suitable $W$-generic $G$. For a given model $V$, the collection of all of its ground models forms the context for what we call set-theoretic geology. This second-order collection, consisting of (possibly many) proper classes $W$, nonetheless admits a first-order definition – within a single universe, we have first-order access to an interesting local neighborhood of the set-theoretic multiverse. We will explore this neighborhood, pointing out various geological phenomena including bedrock models, the mantle and the outer core. This is joint work with Joel David Hamkins and Gunter Fuchs.
Jonas Reitz is an associate professor of mathematics at the CUNY New York City College of Technology. He received his PhD in 2006 under the supervision of Joel David Hamkins. His research interests include forcing, the set-theoretic multiverse, and set-theoretic geology.