 The Recursion Theorem is sometimes also referred to as the Fixed Point Theorem. Note, however, that  Theorem 3.5  does not guarantee the existence of an extensional fixed point for the given function \(f(x)\)—i.e., a number \(n\) such that \(f(n) = n\). (In fact it is evident that there are computable functions for which no such value exists—e.g., \(f(x) = x+1\).) But suppose we view \(f(x)\) instead as a mapping on indices to partial computable functions or, more figuratively, as a means of transforming a program for computing a partial computable function into another program. On this interpretation, the theorem expresses that for every such computable transformation there is some program \(n\) such that the function \(\phi_n(y)\) which it computes is the same as the function \(\phi_{f(n)}(y)\) computed by its image \(f(n)\) under the transformation.