Upward closure and amalgamation in the generic multiverse of a countable model of set theory

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 $d$.

Based on my talk at the conference, Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015. 14 pages. Commentary can be made on my blog at http://jdh.hamkins.org/upward-closure-and-amalgamation-in-the-generic-multiverse

References in corpus (1)

• A multiverse perspective on the axiom of constructiblity

Cited by in corpus (4)

• Cichoń's Diagram and Localisation Cardinals
• Set-Theoretic Blockchains
• The Structure of Models of Second-order Set Theories
• Infinite forcing and the generic multiverse