Incomparable ω1-like models of set theory

We show that the analogues of the Hamkins embedding theo- rems (Ham13), proved for the countable models of set theory, do not hold when extended to the uncountable realm of !1-like models of set theory. Specifically, under the } hypothesis and suitable consistency assumptions, we show that there is a family of 2 !1 many !1-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive !1-like model of ZFC that does not embed into its own constructible universe; and there can be an !1-like model of PA whose structure of hereditarily finite sets is not universal for the !1-like models of set theory.