Hereditary Property - in Set Theory

In Set Theory

Recursive definitions using the adjective "hereditary" are often encountered in set theory.

A set is said to be hereditary (or pure) if all of its elements are hereditary sets. 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, and recursively so is, for example. 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.

A couple of notions are defined analogously:

• A hereditarily finite set is defined as a finite set consisting of zero or more hereditarily finite sets. Equivalently, a set is hereditarily finite if and only if its transitive closure is finite.
• A hereditarily countable set is a countable set of hereditarily countable sets. Assuming the axiom of countable choice, then a set is hereditarily countable if and only if its transitive closure is countable.

Based on the above, it follows that in ZFC a more general notion can be defined for any predicate . A set x is said to have hereditarily the property if x itself and all members of its transitive closure satisfy, i.e. . Equivalently, x hereditarily satisfies iff it is a member of a transitive subset of . A property (of a set) is thus said to be hereditary if is inherited by every subset. For example, being well-ordered is a hereditary property, and so it being finite.

If we instantiate in the above schema with "x has cardinality less than κ", we obtain the more general notion of a set being hereditarily of cardinality less than κ, usually denoted by or . We regain the two simple notions we introduced above as being the set of hereditarily finite sets and being the set of hereditarily countable sets. ( is the first uncountable ordinal.)