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Equivalence of bar recursors in the theory of functionals of finite type

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Abstract

The main result of this paper is the equivalence of several definition schemas of bar recursion occurring in the literature on functionals of finite type. We present the theory of functionals of finite type, in [T] denoted byqf-WE-HA ω, which is necessary for giving the equivalence proofs. Moreover we prove two results on this theory that cannot be found in the literature, namely the deduction theorem and a derivation of Spector's rule of extensionality from [S]: ifP→T 1=T 2 and Q[X∶≡T1], then P→Q[X∶≡ T2], from the at first sight weaker rule obtained by omitting “P→”.

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References

  1. Barendregt, H.P.: The lambda calculus. Amsterdam: North-Holland 1984

    Google Scholar 

  2. Bezem, M.A.: Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals. J. Symb. Logic50, 652–660 (1985)

    Google Scholar 

  3. Curry, H.B., Feys, R.: Combinatory logic. Amsterdam: North-Holland 1958

    Google Scholar 

  4. Howard, W.A.: Functional interpretation of bar induction by bar recursion. Compos. Math.20, 107–124 (1968)

    Google Scholar 

  5. Luckhardt, H.: Extensional Gödel functional interpretation. (Lect. Notes Math., vol. 306) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  6. Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Dekker, J.C.E. (ed.): Proceedings of symposia in pure mathematics V, pp. 1–27. Providence: AMS 1962

    Google Scholar 

  7. Tait, W.W.: Normal form theorem for bar recursive functions of finite type. In: Proceedings of the second scandinavian logic symposium, pp. 353–367. Amsterdam: North-Holland 1971

    Google Scholar 

  8. Troelstra, A.S. (ed.): Metamathematical investigation of intuitionistic arithmetic and analysis. (Lect. Notes Math., vol. 344) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  9. Vogel, H.: Ein starker Normalisationssatz für die barrekursiven Funktionale. Arch. Math. Logik Grundlagenforsch.18, 81–84 (1976)

    Google Scholar 

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  1. Centre for Mathematics and Computer Science, P.O. Box 4079, NL-1009, AB Amsterdam, The Netherlands

    Marc Bezem

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  1. Marc Bezem
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Bezem, M. Equivalence of bar recursors in the theory of functionals of finite type. Arch Math Logic 27, 149–160 (1988). https://doi.org/10.1007/BF01620763

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