# Does the beth function have fixed points of arbitrarily large cofinality?

## 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 $f : \beta \to \alpha$.

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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 $\lambda$.

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