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

TL;DR

This paper proves that adding the mma-rule to the mbda-calculus results in a highly complex logical theory, specifically mbda_{1}^{1}-complete, resolving a long-standing open problem in the field.

Contribution

It establishes the exact logical complexity of the mma-rule in the mbda-calculus as mbda_{1}^{1}-complete, answering a question posed by Barendregt in 1975.

Findings

The mma-rule in the mbda-calculus is mbda_{1}^{1}-complete.

The result characterizes the logical strength of the mma-rule.

The paper resolves a 50-year-old open problem in lambda calculus theory.

Abstract

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.