Abstract
We define a fragment 2λ(Peano) of Girard’s second order λ-calculus 2λ, in which arguments of type application are required to be “Peano types”, namely types generated from type variables and types of the form \( \forall \vec{R}.\tau \) (τ quantifier free of rank ≤ 2) using → and substitution. We show that the provably recursive functions of first order arithmetic are precisely the numeric functions that are λ-definable in 2λ(Peano).
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Leivant, D. (2001). Peano’s Lambda Calculus: The Functional Abstraction Implicit in Arithmetic. In: Anderson, C.A., Zelëny, M. (eds) Logic, Meaning and Computation. Synthese Library, vol 305. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0526-5_14
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DOI: https://doi.org/10.1007/978-94-010-0526-5_14
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