4 - Recursive Enumerability

Abstract

The class of general recursive partial functions is exactly the same as the class of register-machine computable partial functions. The fact that two different approaches yield the same class of functions is evidence that one has here a “natural” class. The members of this class are called computable partial functions. It covers both the total and nontotal functions; it can be omitted in cases where one knows that the function is total. Church's thesis is the assertion that the concept of being a computable partial function is the correct formalization of the informal idea of being an effectively calculable partial function. The class of computable partial functions includes all of the primitive recursive functions. A partial function is a computable partial function if and only if its graph is a recursively enumerable relation.