Background
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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](http://en.wikipedia.org/wiki/Fixed-point_lemma_for_normal_functions). 


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



Does the beth function have fixed points of arbitrarily large cofinality?
Yes. The definable class $C$ of fixed points $\beth\_\kappa = \kappa$ is closed and unbounded and therefore contains ordinals of all possible cofinalities. Specifically, let $\kappa\_\alpha$ denote the $\alpha$-th element of $C$. Note that $\kappa\_\delta = \sup\_{\alpha<\delta} \kappa\_\alpha$ for every limit ordinal $\delta$. If $\lambda$ is a regular cardinal then $\kappa\_\lambda$ has cofinality $\lambda$.


