Abstract.
The paper builds on a simply typed term system \({\cal PR}^\omega \) providing a notion of partial primitive recursive functional on arbitrary Scott domains \(D_\sigma\) that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension \({\cal PR}^{\omega e}\) is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist are certain type 1 Gödel recursors \({\cal R}_k\) for simultaneous partial primitive recursion. Formally, the value \({\cal R}_k\vec{g}\vec{H} x i\) is a function \(f_i\) of type \(\iota \to \iota\), however, \({\cal R}_k\) is modelled such that \(f_i\) is a finite element of \(D_{\iota\to\iota}\) or in other words, a partial sequence. It is shown that the Ackermann function is not definable in \({\cal PR}^{\omega e}\).
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Received July 27, 1995
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Niggl, KH. Non-definability of the Ackermann function with type 1 partial primitive recursion. Arch Math Logic 37, 1–13 (1997). https://doi.org/10.1007/s001530050077
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DOI: https://doi.org/10.1007/s001530050077