How prove the general recursion theorem

Question

Theorem

Let be $A$ a set and let be $S:=\bigcup_{n\in\Bbb{N}}(A^n)$ and finally let be $g:S\rightarrow A$ a function. So there exist a unique function $f:\Bbb{N}\rightarrow A$ such that $$f(n)=g(f|_n)$$ for any $n\in\Bbb{N}$.

My text suggest to use the recursion to prove the last theorem: however I can't prove it and so could someone help me, please?

Answer 1

Define $\varphi:S\to S$ as follows: if $\sigma\in A^n$, then $\varphi(\sigma)=\sigma\cup\{\langle n,g(\sigma)\rangle\}$. If you think of $\sigma$ as a sequence $\langle\sigma(0),\sigma(1),\ldots,\sigma(n-1)\rangle$ of element of $A$, $\varphi$ simply extends that sequence by one term, so that $\varphi(\sigma)$ can be thought of as the sequence $\langle\sigma(0),\sigma(1),\ldots,\sigma(n-1),g(\sigma)\rangle$. By the recursion theorem there is a function $\Phi:\Bbb N\to S$ such that $\Phi(0)=\varnothing$ and $\Phi(n+1)=\varphi(\Phi(n))$ for each $n\in\Bbb N$.

• Prove by induction that $\Phi(n)\in A^n$ for each $n\in\Bbb N$.
• Prove that $\Phi(n+1)\upharpoonright n=\Phi(n)$ for each $n\in\Bbb N$.

Define $f:\Bbb N\to A:n\mapsto\big(\Phi(n+1)\big)(n)$.

• Show by induction that $f\upharpoonright n=\Phi(n)$ for each $n\in\Bbb N$.
• Show that $f(n)=g(f\upharpoonright n)$ for each $n\in\Bbb N$.