This will be a talk for the conference Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) in Kyoto, Japan, September 16-18, 2015.

Abstract. Consider a countable model of set theory amongst its forcing extensions, the ground models of those extensions, the extensions of those models and so on, closing under the operations of forcing extension and ground model.  This collection is known as the generic multiverse of the original model.  I shall present a number of upward-oriented closure results in this context. For example, for a long-known negative result, it is a fun exercise to construct forcing extensions $M[c]$ and $M[d]$ of a given countable model of set theory $M$, each by adding an $M$-generic Cohen real, which cannot be amalgamated, in the sense that there is no common extension model $N$ that contains both $M[c]$ and $M[d]$ and has the same ordinals as $M$. On the positive side, however, any increasing sequence of extensions $M[G_0]\subset M[G_1]\subset M[G_2]\subset\cdots$, by forcing of uniformly bounded size in $M$, has an upper bound in a single forcing extension $M[G]$. (Note that one cannot generally have the sequence $\langle G_n\mid n<\omega\rangle$ in $M[G]$, so a naive approach to this will fail.)  I shall discuss these and related results, many of which appear in the “brief upward glance” section of my recent paper:  G. Fuchs, J. D. Hamkins and J. Reitz, Set-theoretic geology.