The Omega Rule is mathbf{Pi₁¹}-Complete in the λβ-Calculus

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.