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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 a
_{1},...,a_{k}are hereditarily finite, then so is {a_{1},...,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 V_{0}, then V_{1} = P(V_{0}), V_{2} = P(V_{1}),..., V_{k} = P(V_{k−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 V_{n} is finite for any finite n (its cardinality is ^{n−1}2, 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.

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Last updated on Wednesday April 09, 2008 at 09:13:46 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday April 09, 2008 at 09:13:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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