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In [12], Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for $\alpha < {\varepsilon _0}$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Ti then terms built from $BR^{\mathbb{N},\sigma } $ for this particular Y are definable in the fragment ${T_{i + {\rm{max}}\left\{ {1,{\rm{level}}\left( \sigma \right)} \right\} + 2}}$ .

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[1]Berardi, S., Oliva, P., and Steila, S., An analysis of the Podelski-Rybalchenko termination theorem via bar recursion. Journal of Logic and Computation, (2015), First published online: August 28, 2015. Available at
[2]Diller, J., Zur Theorie rekursiver Funktionale höherer Typen, Habilitationsschrift, München, 1968.
[3]Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, vol. 12 (1958), pp. 280287.
[4]Howard, W. A., Functional interpretation of bar induction by bar recursion. Compositio Mathematica, vol. 20 (1968), pp. 107124.
[5]Howard, W. A., Ordinal analysis of bar recursion of type zero. Compositio Mathematica, vol. 42 (1980), no. 1, pp. 105119.
[6]Howard, W. A., Ordinal analysis of simple cases of bar recursion, this Journal, vol. 46 (1981), no. 1, pp. 17–30.
[7]Kohlenbach, U., On the no-counterexample interpretation, this Journal, vol. 64 (1999), no. 4, pp. 1491–1511.
[8]Kreuzer, A., Primitive recursion and the chain antichain principle. Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 2, pp. 245265.
[9]Podelski, A. and Rybalchenko, A., Transition invariants, Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, 2004, pp. 3241.
[10]Ramsey, F. P., On a problem in formal logic. Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264286.
[11]Schwichtenberg, H., Elimination of higher type levels in definitions of primitive recursive functionals by means of transfinite recursion, Proceedings of the Logic Colloquium’73 (Rose, H. E. and Shepherdson, J. C., editors), Studies in Logic and the Foundations of Mathematics, vol. 80, Elsevier, New York, 1975, pp. 279303.
[12]Schwichtenberg, H., On bar recursion of types 0 and 1, this Journal, vol. 44 (1979), no. 3, pp. 325–329.
[13]Spector, C., Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles in current intuitionistic mathematics, Recursive Function Theory (Dekker, F. D. E., editor), Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1962, pp. 127.
[14]Tait, W. W., Infinitely long terms of transfinite type, Formal Systems and Recursive Functions (Crossley, J. N. and Dummett, M. A. E., editors), Studies in Logic and the Foundations of Mathematics, vol. 40, Elsevier, Amsterdam, 1965, pp. 176185.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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