Incomparable $ω_1$-like models of set theory
Fuchs, Gunter, Victoria Gitman, Joel David Hamkins · arXiv (Cornell University) · 2015
We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $ω_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{ω_1}$ many $ω_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $ω_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $ω_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $ω_1$-like models of set theory.