Abstract. I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.
This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.
- My talk at RIMS: Upward closure in the generic multiverse of a countable model of set theory
- My talk at CTFM: Universality and embeddability amongst the models of set theory
- G. Fuchs, J. D. Hamkins, J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015.
- Upward closure in the toy multiverse of all countable models of set theory
- Upward countable closure in the generic multiverse of forcing to add a Cohen real