Aleph number

Summary

The aleph numbers $\aleph_\alpha$, indexed by the ordinals, are the infinite cardinal numbers of well-orderable sets, listed in increasing order. The first is aleph-null, the size of the natural numbers, and $\aleph_1$ is the next infinite cardinal. With the axiom of choice, every infinite set has an aleph as its cardinality, so the alephs measure all infinite sizes. The precise definition is by transfinite recursion.