beth numbersBethNumbers2004-04-02 07:31:522006-12-30 03:14:47Definitioncardinal number\usepackage{amssymb}
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The \emph{beth numbers} are infinite cardinal numbers
defined in a similar manner to the aleph numbers, as described below.
They are written $\beth_\alpha$, where $\beth$ is beth,
the second letter of the Hebrew alphabet,
and $\alpha$ is an ordinal number.
We define $\beth_0$ to be the first infinite cardinal (that is, $\aleph_0$).
For each ordinal $\alpha$,
we define $\beth_{\alpha+1}=2^{\beth_\alpha}$.
For each limit ordinal $\delta$,
we define $\beth_\delta=\bigcup_{\alpha\in\delta}\beth_\alpha$.
Note that $\beth_1$ is the cardinality of the continuum.
For any ordinal $\alpha$ the inequality $\aleph_\alpha\leq\beth_\alpha$ holds.
The Generalized Continuum Hypothesis is equivalent to the assertion that
$\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$.
For every limit ordinal $\delta$,
the cardinal $\beth_\delta$ is a strong limit cardinal.
Every uncountable strong limit cardinal arises in this way.