Anyway, we expect that this obtaining back information can be done in an effective way, by an appropriate total recursive function.

The set S is the range of a partial recursive function.

The set S is the range of a total recursive function or empty.

This recursive function terminates if either conditions 1 or 2 are satisfied.

How is a recursive function written in these languages?

In this setting a term is a primitive recursive function of zero or more variables.

But there is no reason, in general, to suppose that such a recursive function exists.

The set of all recursive functions is known as R in computational complexity theory.

The reader is probably unfamiliar with the notion of a "recursive function".

Thus most of life goes on requiring only the "primitive recursive functions."