The concept of computable function was at first given intuitively (§ 2). We have, by virtue of an analysis of the behaviour of a calculator (§ 3), arrived at an exact definition of Turing-computability (§ 6). The direct connection with intuition, which is gained by this method, is without doubt a great advantage in realizing the meaning of the precise concepts obtained. On the other hand, the concept of Turing-computability, just as it stands, is not flexible enough for the work of the mathematician. When we want to consider the properties of computable functions we shall try to find, as all mathematicians would do, a new, to the original equivalent definition, which can more easily be handled mathematically. We know today several concepts which are equivalent to Turing-computability. Each one of these new concepts has an intuitive background as well. However, this background is on the whole not of the kind that we would be inclined to believe relatively quickly (as in the case of Turing-computability) that the precise replacement obtained on such a basis comprehends all possible computable functions. The fact that for every one of these concepts we can prove rigorously the equivalence to Turing-computability strengthens in any case the conviction that in all these investigations we are dealing with a quite fundamental concept.
KeywordsInitial Function Recursive Function Computable Function Induction Schema Number Pair
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- Kleene, S. C.: General Recursive Functions of Natural Numbers. Math. Ann. 112, 727–742 (1936).Google Scholar