A characterization of the $\Sigma_1$-definable functions of $KP\omega + (uniform\; AC)$

Abstract.

The subject of this paper is a characterization of the $\Sigma_1$-definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$. This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$. This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.