proof of existence in the recursion theorem in set theory

Question

consider this theorem of set theory, the recursion theorem. On Wikipedia there is a proof of uniqueness but no proof of existence, I would like to see an existence proof.

Answer 1

You can show by induction that for any $n$, there is a unique function $F_n$ with domain $n+1$ that satisfies the recursion equations. The base case is $F_0 = \{(0,a)\}$ and then $F_{n+1}=F_n\cup\{(n+1,f(F_n(n))\}$ satisfies the equations provided $F_n$ does, and is unique by the same proof as the uniqueness proof on wikipedia. Then take $F=\bigcup_{n\in\mathbb N}F_n$ and prove it has the required properties.