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arXiv · 1511.01074

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

Joel David Hamkins
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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 $d$.

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Joel David Hamkins. 2015-11-03. Upward closure and amalgamation in the generic multiverse of a countable model of set theory. https://arxiv.org/abs/1511.01074

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