UPWARD CLOSURE AND AMALGAMATION IN THE GENERIC MULTIVERSE OF A COUNTABLE MODEL OF SET THEORY (Recent Developments in Axiomatic Set Theory)
Joel David Hamkins · Kyoto University Research Information Repository (Kyoto University) · 2016
I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable tran- sitive 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 forc- ing 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 $d.$Consider a countable transitive model of set theory $W\models$ ZFCin the context of all its forcing extensions.Several natural questions im- mediately suggest themselves concerning issues of amalgamation and upward-closure.For example, can any two such models be amalga- mated into a common larger model?In other words, is this collection of models upward directed?When can we expect to find upper bounds for increasing chains?In this article, I shall resolve these and other similar This article is based upon I talk I gave at the conference on Recent Develop- ments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto