  We show that for any $k\in\omega$, the structure $(H_k,\in)$ of sets that are
hereditarily of size at most $k$ is decidable. We provide a transparent
complete axiomatization of its theory, a quantifier elimination result, and
tight bounds on its computational complexity. This stands in stark contrast to
the structure $V_\omega=\bigcup_k H_k$ of hereditarily finite sets, which is
well known to be bi-interpretable with the standard model of arithmetic
$(\mathbb N,+,\cdot)$.
