Some Set-Theoretic Reduction Principles

In this article we study several reduction principles in the context of Simpson’s set theory and Kripke-Platek set theory KP (with infinity). Since is the set-theoretic version of ATR0 there is a direct link to second order arithmetic and the results for reductions over are as expected and more or less straightforward. However, over KP we obtain several interesting new results and are lead to some open questions.