aleph numbers AlephNumbers 2004-03-15 10:54:22 2006-12-30 03:14:32 Definition cardinal number \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\N{\mathbb{N}} \def\Q{\mathbb{Q}} \def\Z{\mathbb{Z}} \PMlinkescapeword{alphabet} The \emph{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 \PMlinkname{Well-Ordering Principle}{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 $\kappa=\aleph_\alpha$.