 Hereditarily countable set

A set is hereditarily countable when its transitive closure is countable. These sets form the transitive set $H_{{\omega }_1}$ and all have rank below ${\omega }_1$.

A set is called **hereditarily countable** if it is countable, and all of its elements (and their elements, recursively) are also countable. In more formal terms, a set \( A \) is hereditarily countable if: 1. \( A \) is countable. 2. Every element of \( A \) is countable. 3. Every element of every element of \( A \) is countable, and so on.