[39] Let f be any computable function. Prove that there exists an n such that n = f(n). Comments. This result, and its elaborations to

[39] Let f be any computable function. Prove that there exists an n such that φn = φf(n).

Comments. This result, and its elaborations to more complicated versions, is usually called the second recursion theorem or the fixed-point theorem for computability theory. The n is called a fixed-point value for

f. Standard applications include the following: There exists an e such that the only element in the domain of φe is e itself, and more generally, it allows us to write programs that know their own index. Source: [H.

Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967].