In my current set-theoretical endeavors, I have stumbled upon this proof of the recursion theorem. However, there's something fundamental I don't get here: First, in the definition of $A(n)$ (first few lines of the proof), the author writes $\forall m< n: h(s(n)) = g(h(m))$. However, when it comes to the inductive step, the author writes
To check that $h'\in A(s(n)$, we have to verify that $\forall m< n: h'(s(m)) = g(h'(m)$.
I find this confusing for 2 reasons:
- Wouldn't the author have to verify $h'(s(s(n))) = g(h'(m)$ instead? You know, just substituting n for s(n) in the inductive step.
- Even if my assumption is wrong - why is he writing the m's inside the brackets when writing $h'(s(m)) = g(h'(m)$ ? This is not what he is presupposing A(n) to fulfill when defining it initially as far as I understand.
There is an error in the definition of $A(n)$ in the proof: the equation in the definition should be $h(s(m))=g(h(m))$, not $h(s(n))=g(h(m))$. With this correction the induction step should make more sense.