Abstract

In this article we study several reduction principles in the context of Simpson’s set theory ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} and Kripke-Platek set theory KP (with infinity). Since ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} is the set-theoretic version of ATR0 there is a direct link to second order arithmetic and the results for reductions over ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} are as expected and more or less straightforward. However, over KP we obtain several interesting new results and are lead to some open questions.