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arXiv · 0903.1374

The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus

Benedetto Intrigila·Richard Statman
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Abstract

In a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term N 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.

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Benedetto Intrigila, Richard Statman. 2009-03-07. The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus. https://doi.org/10.2168/lmcs-5(2:6)2009

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