Incomparable $ω_1$-like models of set theory

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.

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