Journal of Symbolic Logic

The Role of Parameters in Bar Rule and Bar Induction

Michael Rathjen

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Abstract

For several subsystems of second order arithmetic $\mathbf{T}$ we show that the proof-theoretic strength of $\mathbf{T}$ + (bar rule) can be characterized in terms of $\mathbf{T} +$ (bar induction)$^\square$, 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 $\mathbf{ACA}^+_0, \mathbf{ACA}_0 +$ (bar rule) and $\mathbf{ACA}_0 +$ (bar induction)$^\square$ prove the same $\Pi^1_1$-sentences.

Article information

Source
J. Symbolic Logic, Volume 56, Issue 2 (1991), 715-730.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743670

Mathematical Reviews number (MathSciNet)
MR1133098

Zentralblatt MATH identifier
0736.03024

JSTOR
links.jstor.org

Citation

Rathjen, Michael. The Role of Parameters in Bar Rule and Bar Induction. J. Symbolic Logic 56 (1991), no. 2, 715--730. https://projecteuclid.org/euclid.jsl/1183743670


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