The role of parameters in bar rule and bar induction

Journal of Symbolic Logic 56 (2):715-730 (1991)
Abstract
For several subsystems of second order arithmetic T we show that the proof-theoretic strength of T + (bar rule) can be characterized in terms of T + (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 ACA + 0 , ACA 0 + (bar rule) and ACA 0 + (bar induction) □ prove the same Π 1 1 -sentences
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DOI 10.2307/2274713
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