Hereditarily countable set

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In set theory, a set is called hereditarily countable if and only if it is a countable set of hereditarily countable sets. This inductive definition is in fact well-founded and can be expressed in the language of first-order set theory. A set is hereditarily countable if and only if every element of its transitive closure is countable. If the axiom of countable choice holds, then a set is hereditarily countable if and only if its transitive closure is countable.

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, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.