Hereditary set

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In the mathematical field of 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.

In most standard formulations of set theory, intended to be interpreted in the von Neumann universe or to express the content of Zermelo-Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there may be urelements.