  In order to build the collection of Cauchy reals as a set in constructive set
theory, the only Power Set-like principle needed is Exponentiation. In
contrast, the proof that the Dedekind reals form a set has seemed to require
more than that. The main purpose here is to show that Exponentiation alone does
not suffice for the latter, by furnishing a Kripke model of constructive set
theory, CZF with Subset Collection replaced by Exponentiation, in which the
Cauchy reals form a set while the Dedekind reals constitute a proper class.
