Hereditarily finite set

A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a proper class. The set in question is \(V_\omega = L_\omega\) in the cumulative/constructible hierarchies, while there is no \(\alpha\) so that \(V_\alpha\) or \(L_\alpha\) contains all the finite sets.