Gabriel's Horn

$~V_{4}~$ represented with circles in place of curly brackets    

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