Nested set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

A recursive definition of well-founded hereditarily finite sets goes as follows:

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

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 ${\displaystyle H_{\aleph _{0}}}$, meaning hereditarily of cardinality less than ${\displaystyle \aleph _{0}}$.

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