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
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
See also
- Hereditarily countable set
- Hereditarily finite set
- Well-founded set
References
- ISBN 0-444-85401-0.