Gödel’s theorem

We will introduce a class of numerical functions which evidently are effectively computable. The procedure may seem rather ad hoc, but it gives us a surprisingly rich class of algorithms. We use the inductive method, that is, we fix a number of initial functions which are as effective as one can wish; after that we specify certain ways to manufacture new algorithms out of old ones.

The initial algorithms are extremely simple indeed: the successor function, the constant functions and the projection functions. It is obvious that composition (or substitution) of algorithms yields algorithms The use of recursion was as a device to obtain new functions already known to Dedekind'; that recursion produces algorithms from given algorithms is also easily seen. In logic the study of primitive recursive functions was initiated by Skolem, Herbrand, Gödel and others.