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.

Source : oai:arXiv.org:0903.1374

DOI : 10.2168/LMCS-5(2:6)2009

Volume: Volume 5, Issue 2

Published on: April 27, 2009

Submitted on: February 21, 2008

Keywords: Computer Science - Logic in Computer Science,F.4.1

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