aleph numbers

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

To start the transfinite recursion,we define ${\mathrm{\aleph }}_0$ to be the first infinite ordinal.This is the cardinality of countably infinite sets, such as $\mathbb{N}$ and $\mathbb{Q}$.For each ordinal $\alpha$,the cardinal number ${\mathrm{\aleph }}_{\alpha +1}$ is defined to bethe least ordinal of cardinality greater than ${\mathrm{\aleph }}_{\alpha }$.For each limit ordinal $\delta$,we define ${\mathrm{\aleph }}_{\delta }={\bigcup }_{\alpha \in \delta }{\mathrm{\aleph }}_{\alpha }$.

As a consequence of the Well-Ordering Principle (http://planetmath.org/ZermelosWellOrderingTheorem),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 $\kappa ={\mathrm{\aleph }}_{\alpha }$.