  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 $\omega_1$-like models of set theory. Specifically, under the
$\diamondsuit$ hypothesis and suitable consistency assumptions, we show that
there is a family of $2^{\omega_1}$ many $\omega_1$-like models of ZFC, all
with the same ordinals, that are pairwise incomparable under embeddability;
there can be a transitive $\omega_1$-like model of ZFC that does not embed into
its own constructible universe; and there can be an $\omega_1$-like model of PA
whose structure of hereditarily finite sets is not universal for the
$\omega_1$-like models of set theory.
