## Journal of Symbolic Logic

### The Role of Parameters in Bar Rule and Bar Induction

Michael Rathjen

#### 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

https://projecteuclid.org/euclid.jsl/1183743670

Mathematical Reviews number (MathSciNet)
MR1133098

Zentralblatt MATH identifier
0736.03024

JSTOR