## Benedetto Intrigila ; Richard Statman - The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus

lmcs:1147 - Logical Methods in Computer Science, April 27, 2009, Volume 5, Issue 2 - https://doi.org/10.2168/LMCS-5(2:6)2009
The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus

Authors: Benedetto Intrigila ; Richard Statman

In a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term <i>N</i> return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the \lambda\beta-calculus the \Omega-rule does not hold, even when the \eta-rule (weak extensionality) is added to the calculus. A long-standing problem of H. Barendregt (1975) concerns the determination of the logical power of the \Omega-rule when added to the \lambda\beta-calculus. In this paper we solve the problem, by showing that the resulting theory is \Pi\_{1}^{1}-complete.

Volume: Volume 5, Issue 2
Published on: April 27, 2009
Accepted on: June 25, 2015
Submitted on: February 21, 2008
Keywords: Computer Science - Logic in Computer Science,F.4.1

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