The role of parameters in bar rule and bar induction

Abstract

For several subsystems of second order arithmeticTwe show that the proof-theoretic strength ofT+ (bar rule) can be characterized in terms ofT+ (bar induction)^□, where the latter scheme arises from the scheme of bar induction by restricting it to well-orderings with no parameters. In addition, we demonstrate that,ACA₀+ (bar rule) andACA₀+ (bar induction)^□prove the same-sentences.