# Hereditary set

In set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on. For example, it is vacuously true that the empty set is a hereditary set, and thus the set $\{\varnothing\}$ containing only the empty set $\varnothing$ is a hereditary set.