The aleph numbers are infinite cardinal numbers defined by transfinite recursion, as described below. They are written $\aleph_\alpha$ where $\aleph$ is aleph, the first letter of the Hebrew alphabet, and $\alpha$ is an ordinal number. Sometimes we write $\omega_\alpha$ instead of $\aleph_\alpha$ usually to emphasise that it is an ordinal.

To start the transfinite recursion, we define $\aleph_0$ to be the first infinite ordinal. This is the cardinality of countably infinite sets, such as $\N$ and $\Q$ For each ordinal $\alpha$ the cardinal number $\aleph_{\alpha+1}$ is defined to be the least ordinal of cardinality greater than $\aleph_\alpha$ For each limit ordinal $\delta$ we define $\aleph_\delta=\bigcup_{\alpha\in\delta}\aleph_\alpha$

As a consequence of the Well-Ordering Principle, every infinite set is equinumerous with an aleph number. Every infinite cardinal is therefore an aleph. More precisely, for every infinite cardinal $\kappa$ there is exactly one ordinal $\alpha$ such that $\kappa=\aleph_\alpha$