Hereditarily finite sets

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,...,ak are hereditarily finite, then so is {a1,...,ak}.

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

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 Vn 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. Vω is also symbolized by H_{aleph_0}, meaning hereditarily of cardinality less than aleph_0. See also hereditarily countable set.

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