 The set theory KPU (Kripke-Platek set theory with urelements, for which see Barwise 1975) is of interest for technical reasons in model theory. The axioms of KPU are the weak Extensionality which allows urelements, Pairing, Union, \(\Delta_0\) separation, \(\Delta_0\) collection, and \(\in\)-induction for arbitrary formulas. Note the absence of Power Set. The technical advantage of KPU is that all of its constructions are “absolute” in a suitable sense. This makes the theory suitable for the development of an extension of recursion theory to sets.