In computer science and engineering, a system acts as a computable function.

The class of computable functions that return 0 for every input, and its complement.

The class of computable functions that are constant, and its complement.

Let be a set of computable functions such that .

This term has since come to be identified with the computable functions.

The greatest common divisor of two numbers is a computable function.

R is equal to the set of all total computable functions.

Each computable function has an infinite number of different program representations in a given programming language.

To complete the proof, let be any total computable function, and construct as above.

Although Kleene did not give examples of "computable functions" others have.