Beth number

Summary

The beth numbers $\beth_\alpha$ are the infinite cardinal numbers obtained from $\aleph_0$ by taking power sets again and again, and suprema at limit stages. So $\beth_0 = \aleph_0$, and $\beth_1 = 2^{\aleph_0}$ is the cardinality of the real numbers. The precise definition is by transfinite recursion. The generalized continuum hypothesis says exactly that $\beth_\alpha = \aleph_\alpha$ for every ordinal $\alpha$.