Hereditarily finite set

In mathematics, hereditarily finite sets are defined recursively as finite sets containing only hereditarily finite sets (with the empty set as a base case). Informally, a hereditarily finite set is a finite set, the members of which are also finite sets, as are the members of those, and so on.

They are constructed by the following rules:

The empty set is a hereditarily finite set.

If a₁,...,a_k are hereditarily finite, then so is {a₁,...,a_k}.

The set of all hereditarily finite sets is denoted V_ω. If we denote P(S) for the power set of S, V_ω can also be constructed by first taking the empty set written V₀, then V₁ = P(V₀), V₂ = P(V₁),..., V_k = P(V_k−1),... Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup_{k=0}^{\infty} V_k = V_\omega.

The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consisting of the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.

Notice that there are countably many hereditarily finite sets, since V_n is finite for any finite n (its cardinality is ⁿ⁻¹2, see tetration), and the union of countably many finite sets is countable.

Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. V_ω is also symbolized by Failed to parse (Missing texvc executable; please see math/README to configure.): H_{\aleph_0} , meaning hereditarily of cardinality less than Failed to parse (Missing texvc executable; please see math/README to configure.): \aleph_0 . See also hereditarily countable set.