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 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, meaning hereditarily of cardinality less than .
Read more about this topic: Hereditarily Finite Set
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