Aleph number

Remarks

Example: $\aleph_1$ is the set of countable ordinals, which is the least uncountable ordinal $\omega_1$. Check: each countable ordinal is smaller than $\omega_1$, and $\omega_1$ itself is uncountable, so it is the least cardinal above $\aleph_0$. Non-example: $\omega + 1$ is an infinite ordinal but not an aleph, since it has the same cardinality as the smaller ordinal $\omega$. The cardinal $\kappa^+$ exists in ZF by Hartogs' theorem, without the axiom of choice. The map $\alpha \mapsto \aleph_\alpha$ is a normal function, and every infinite well-orderable cardinal is $\aleph_\alpha$ for exactly one $\alpha$. The ordinal $\aleph_\alpha$ is also written $\omega_\alpha$ when it is viewed as an ordinal.