Context: self-study from Smullyan and Fitting's "Set Theory and the Continuum Problem" (revised 2010 edition), chapter 3, section 8, Definition by Finite Recursion.
They give Theorem 8.1
For any function $g$ from $A$ into $A$ and any set $c \in A$ there is a unique function $f$ from numbers to elements of $A$ such that: $$(1): \quad f(0) = c$$ $$(2): \quad \forall n: f(n^+)= g(f(n))$$
By "numbers" here he means $\omega$, the Von Neumann construction which has been proved to be a minimally inductive set, all that good stuff that we know and love. (Warner covers the same area more clumsily by implementing it as a naturally ordered semigroup but gets to the same place via a different path.
$A$ in the above is a general class.
Then we come to Exercise 8.3:
Prove the following strengthening of Theorem 8.1. For any class $A$ and any element $c \in A$ and any function $g(x, y)$ from $\omega \times A$ into $A$, there is a unique function $f$ from $\omega$ into $A$ such that $f(0) = c$ and for any number $n$, $f(n^+) = g(n, f(n))$. Why is this a strengthening of Theorem 8.1?
It's straightforward but tedious to prove the above, once you know what you're doing (as I say, Warner was a fair model for that). But I'm at a loss to know what they mean by "Why is this a strengthening of Theorem 8.1?"
I believe he means you can use the strengthened version to prove Theorem 8.1 directly, but Ican't see how to convert $g(n, f(n))$ into $g(f(n))$ and make sense of it.
Is this all he means? To use the strengthened version to prove the unstrengthened version? If so, how do you go about it?
Or is it that having proved the strengthened version, you can then (via induction on the number of parameters) deduce the general proof by finite recurrence by positing a function $g$ with an arbitrary arity?
Yes all right I get the message, stupid question, I'll do my own answer.
Let $h: A \to A$ be the mapping: $$\forall x \in A: h (x) = g (a, x)$$ where $a$ is an arbitrary element of $\omega$.
Then: $$\forall y \in \omega: \forall x \in A: h(x) = g(y, x)$$
Then you can just plug $h(f(n))$ into the $g(n, f(n))$ in the exercise, and that directly leads to Theorem $8.1$.
Downvote into oblivion if I've completely misunderstood the barest fundamentals of this subject.