  It is quite easy to see that if A is a transitive set such that &lt;A, ∈ | A&gt; is a model of ZFC, then A is admissible. More generally, the result continues to hold when the power set axiom is omitted from ZFC, so that both H(ω) and H(ω1) are admissible. However, since the latter is uncountable, the Barwise compactness theorem fails to apply to it.