Hereditarily finite set

"Nested set" redirects here. Nested set may also refer to the Nested set model in relational databases.

In mathematics and set theory, hereditarily finite sets are defined recursively as finite sets consisting of 0 or more hereditarily finite sets.

Formal definition

A recursive definition of a hereditarily finite set goes as follows:

Base case: The empty set is a hereditarily finite set.

Recursion rule: 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

$\bigcup_{k=0}^\infty V_k = V_\omega.$

Discussion

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 $H_{\aleph_0}$, meaning hereditarily of cardinality less than $\aleph_0$.

See also

• Hereditarily countable set