# Hereditary set

In set theory, a **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.

## Examples

For example, it is vacuously true that the empty set is a hereditary set, and thus the set containing only the empty set is a hereditary set.

## In formulations of set theory

In formulations of set theory that are 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.

## Assumptions

The inductive definition of hereditary sets presupposes that set membership is well-founded (i.e., the axiom of regularity), otherwise the recurrence may not have a unique solution. However, it can be restated non-inductively as follows: a set is hereditary if and only if its transitive closure contains only sets. In this way the concept of hereditary sets can also be extended to non-well-founded set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set.

## See also

## References

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