Determine the interleaving of the aleph and beth hierarchies

Determine how the aleph numbers (ℵn, defined by successive cardinals) and the beth numbers (ℶn, defined by iterated power-set cardinalities with ℶ0 = ℵ0 and ℶn+1 = 2^{ℶn}) are interleaved; specifically, characterize the precise relative ordering between these two canonical sequences of infinite cardinals and identify, when possible, which aleph equals a given beth.

Background

The paper introduces two standard hierarchies of infinite cardinals: the aleph sequence (ℵ0, ℵ1, ℵ2, …) obtained by taking successive cardinals, and the beth sequence (ℶ0, ℶ1, ℶ2, …) obtained by iterating the power-set operation, with ℶ0 = ℵ0 and ℶn+1 = 2^{ℶn}.

While each hierarchy is strictly increasing on its own, the author notes that their precise relationship is not determined by the basic axioms discussed. Establishing exactly how these two sequences are interleaved would clarify which alephs coincide with which beths and in what order, a question tied to well-known issues such as the (generalized) continuum hypothesis.