The code is a fixed point of a transformation, that transformation being "running it and extracting the output". Indeed, the fact that they always exist is a s

The code is a fixed point of a transformation, that transformation being "running it and extracting the output". Indeed, the fact that they always exist is a specialization of a more general fixed-point theorem! See: https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem