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

To start the transfinite recursion, we define to be the first infinite ordinal. This is the cardinality of countably infinite sets, such as and . For each ordinal , the cardinal number is defined to be the least ordinal of cardinality greater than . For each limit ordinal , we define .

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 there is exactly one ordinal such that .