beth numbers

The 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.