Hereditarily countable set

The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory (ZF) without any form of the axiom of choice, and this set is designated $H_{\aleph_1}$. The hereditarily countable sets form a model of Kripke–Platek set theory with the axiom of infinity (KPI), if the axiom of countable choice is assumed in the metatheory.
If $x \in H_{\aleph_1}$, then $L_{\omega_1}(x) \subset H_{\aleph_1}$.
More generally, a set is hereditarily of cardinality less than κ if and only it is of cardinality less than κ, and all its elements are hereditarily of cardinality less than κ; the class of all such sets can also be proven to be a set from the axioms of ZF, and is designated $H_\kappa \!$. If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.