How is an empty set a hereditary set? I read the definition of a hereditary, It states that " Hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on. " If I look at the empty set, there isn't an element that in there that's a set because it's empty. How does this satisfy the definition of a hereditary set?

Every element of the empty set is a hereditary set. Every element of the empty set is _also_ an elephant. When we universally quantify over the emptyset ("$\forall x\in\emptyset(...)$"), we wind up with a true statement for silly reasons (" **vacuous truth** "); dually, if we existentially quantify over the emptyset ("$\exists x\in\emptyset(...)$") we get a false statement for silly reasons.

The definition of hereditary set isn't _positive_ , it's _negative_ : a set is hereditary if and only if it **doesn't** contain something it **shouldn't** , and the emptyset - containing nothing at all - is therefore hereditary.