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.
KeywordsFunction Variable Recursive Function Computable Function Regular System Finite System
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- Kleene, S. C.: General Recursive Functions of Natural Numbers. Math. Ann. 112, 727–742 (1936).Google Scholar
- 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.)Google Scholar