## Background

The beth function is defined recursively by: $\beth_0 = \aleph_0$, $\beth_{\alpha + 1} = 2^{\beth_\alpha}$, and $\beth_\lambda = \bigcup_{\alpha < \lambda} \beth_\alpha$. Since the beth function is strictly increasing and continuous, it is guaranteed to have arbitrarily large fixed points by the fixed-point theorem on normal functions.

The cofinality of an ordinal $\alpha$ is the smallest ordinal $\beta$ such that there are unbounded increasing functions $f : \beta \to \alpha$.