Incomparable ω 1 ‐like models of set theory

We show that the analogues of the embedding theorems of [3], 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 ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of 2 ω 1many 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.