Abstract
In the last two chapters we considered the properties of μ-recursive functions. It was shown that the class of μ-recursive functions is the same as the class of Turing-computable functions and so the same as the class of the functions which are computable in the intuitive sense. Thus, we can say that the concept of μ-recursive function, just like that of Turing-computable function, is a precise replacement of the concept of computable function. Another concept which can be considered to be a precise replacement of the concept of computable function (and which historically precedes the concept of μ-recursive function) is the concept of recursive function (Herbrand, Gödel, Kleene). After the definition of recursiveness (in § 19) we shall show in the two following paragraphs that the class of μ-recursive functions coincides with the class of recursive functions.
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References
Herbrand, J.: Sur la non-contradiction de l'Arithmétique. J. reine angew. Math. 166, 1–8 (1931). (Idea of recursive function.)
Gödel, K.: On Undecidable Propositions of Formal Mathematical Systems. Mimeographed. Institute for Advanced Study, Princeton, N. J. 1934. 30 pp. (First account of rules of inference.)
Kleene, S. C.: General Recursive Functions of Natural Numbers. Math. Ann. 112, 727–742 (1936). (Introduction of the expression "recursive functions" for these functions, and their precise definition.)
Kleene, S. C.: General Recursive Functions of Natural Numbers. Math. Ann. 112, 727–742 (1936).
Kalmar, L.: Über ein Problem, betreffend die Definition des Begriffes der allgemeinrekursiven Funktion. Z. math. Logik 1, 93–96 (1955). (Here we find the example dealt with in Section 7.)
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© 1969 Springer-Verlag Berlin · Heidelberg
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Hermes, H. (1969). Recursive Functions. In: Enumerability · Decidability Computability. Die Grundlehren der mathematischen Wissenschaften, vol 127. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46178-1_5
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DOI: https://doi.org/10.1007/978-3-642-46178-1_5
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