Beth number

Remarks

Example: $\beth_1 = 2^{\aleph_0}$ is the cardinality of $\R$, since $\R$ is in bijection with the power set of $\N$. $\beth_2 = 2^{2^{\aleph_0}}$ is the cardinality of the power set of $\R$, and also of the set of all functions $\R \to \R$. Non-example: $\aleph_1$ is a beth number only if $\aleph_1 = \beth_1$, which is the continuum hypothesis and is not provable in ZFC. By Cantor's theorem $\beth_{\alpha+1} > \beth_\alpha$, and by induction $\beth_\alpha \ge \aleph_\alpha$ for every $\alpha$. Another notation is $\beth_\alpha = \beth_\alpha(\aleph_0)$; more generally $\beth_0(\kappa) = \kappa$ and $\beth_{\alpha+1}(\kappa) = 2^{\beth_\alpha(\kappa)}$.