Is there computable function to compute each computable real number? Recall that a computable real number is one which can be calculated to any precision, like $\pi$ or $e$. It does not matter that these numbers are irrational, computability is about being able to compute rational approximations.

Is there a computable function which outputs an (infinite) sequence of Turing machines, each of which computes a real number, and such that all real numbers are so included?
There is a real question behind the confusion.

A real number $x$ is said to be computable if there a Turing machine, which upon input $n \in \mathbb{N}$ outputs a pair of numbers $a, b \in \mathbb{N}$ such that $|x - a/b| &lt; 2^{-n}$. That is, there is a machine that computes arbitrarily precise rational approximations of $x$.

We can ask whether there is a machine that outputs an infinite list of machines, such that each of the listed machines computes a real number, and all computable real numbers are so included. That would be the way to make sense of the question.

The answer is negative. There is no such machine. The proof is essentially the diagonalization argument that show that a sequence of reals cannot exhaust the set of reals, except it is modified to work for computable reals and computable sequences.