# 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 (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 $\kappa=\aleph_{\alpha}$.

Title aleph numbers AlephNumbers 2013-03-22 14:15:39 2013-03-22 14:15:39 yark (2760) yark (2760) 6 yark (2760) Definition msc 03E10 alephs GeneralizedContinuumHypothesis BethNumbers