Closure of system T under the bar recursion rule

Paulo Oliva, Queen Mary University of London. Part of the logic seminar series.

Bar recursion (BR) is a form of recursive definition proposed by Spector [1] to extend system T so as to give a computational interpretation to proofs in classical analysis.  It follows that T + BR is strictly more expressive than system T. Bar recursion can be viewed as form of recursion on well-founded trees. A celebrated result of Schwichtenberg [2] implies that if we restrict to T-definable trees, then BR of lowest type will not take us outside T-definability. This closure property is proven in [2] via a reduction to an infinitary version of system T, following by the use of ordinal recursion, and the translation of these back into system T. In [3] we have given a more direct proof, with an explicit construction that given a term in T + BR of restricted type, will convert that into a T term.

[1] Spector, C., Provably recursive functionals of analysis, Recursive Function Theory: Proc. Symposia in Pure Mathematics, 5, 1–27, 1962

[2] Schwichtenberg, H., On bar recursion of types 0 and 1, The Journal of Symbolic Logic, 44, 325–329, 1979

[3] Oliva, P. and Steila, S., A direct proof of Schwichtenberg’s bar recursion closure theorem, The Journal of Symbolic Logic, to appear.