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# aleph numbers

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 $\mathbb{N}$ and $\mathbb{Q}$. 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 $\kappa=\aleph_{\alpha}$.

## Mathematics Subject Classification

03E10*no label found*

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