Hereditarily small set
= Hereditarily small set
{title2=$H_\kappa=\{x:|\operatorname{tc}(\{x\})|<\kappa\}$}
For an infinite <cardinal number> $\kappa$, $H_\kappa$ consists of the sets $x$ for which $|\operatorname{tc}(\{x\})|<\kappa$, where $\operatorname{tc}$ is <transitive closure>. The entire transitive membership ancestry is bounded, not only the size of $x$. These collections are <transitive sets>. At $\kappa=\omega_1$ the elements are the <hereditarily countable sets>.