The theory of hereditarily bounded sets

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 $(\mathbb N,+,\cdot)$.