My understanding of transfinite induction comes from Weaver where he says:
Theorem: Suppose that for every ordinal $\alpha$
$(\forall\beta<\alpha)\phi(\beta)\rightarrow\phi(\alpha)$
holds. Then $(\forall\alpha)\phi(\alpha)$ holds.
I understand this statement - it is a pretty clear extension of "regular" induction - essentially that if knowing $\phi$ is true for all smaller ordinals tell you that $\phi$ is true for the "next" ordinal then $\phi$ is true for all ordinals.
However, Cohen formulation is very different and I'm having trouble following it. He says (note: Cohen also uses $\phi$ in his formulation, but I think it means something different, so I replaced it with $\psi$ to try to avoid confusion):
Definition: let $A_n(x,y,t_1,...t_k)$ run through all formulas in ZF with at least two free variables
Theorem: Let $t_i$ be given and assume $\forall x \exists!yA_n(x,y,t_1,...,t_k)$ so that $A$ defines a function $y=\psi(x)$. For every ordinal $\alpha$ and set $z$ there is a unique function $f$ defined on $\{\beta|\beta\leq\alpha\}$ (this is actually the set $\alpha+1$), such that $f(0)=z$ and for all $\beta\leq\alpha$, $f(\beta)=\psi(h)$ where $h$ is $f$ restricted to $\beta$.
I'm not able to relate these two theorems. I think Weaver's $\phi$ is Cohen's $A_n$ but I'm not sure. Here are some thoughts I've tried:
Thought 1: I've tried to start at $\alpha=0$. Since $0=\emptyset$, then $h$ being $f$ restricted to $0$ is empty i.e. $h=\emptyset$. That means that:
$f(0)=z=\psi(\emptyset)$
But here, I'm already stuck. This is supposed to hold for every $z$ but it can't be true because $\psi(\emptyset)$ is unique.
Thought 2: But, okay, let's say that $f(0)=z$ and the statement $f(\beta)=\psi(h)$ only holds for ordinals greater than $0$. So, then, $f(1)=\phi(h)$ where $h$ is the function mapping $0$ to $z$ (I suppose $h=\{\{0, \{0,h\}\}\}$). So now we know what $f(1)$ is. And we can keep going to $f(2), f(3), ...$ and then $f(\omega)$, etc.... So we've got some function $f$ defined and being "built up" from how $\psi$ "tells us" what to do next (given that we started from $z$). But I still don't see how to relate this back to Weaver's formulation and a way of proving some statement is true for all ordinals.
Any help relating the two, or just understanding what Cohen's formulation is saying, is appreciated.
P.S. before anyone asks, I'm just reading Cohen for fun.
I checked with some friends and got an answer to this myself.
Thought 1: since $f(0) = z$, the statement of the theorem really should be "for all $\beta$ such that $0<\beta\leq\alpha$, $f(\beta)=\psi(h)$".
Thought 2: Cohen's version is really more "Transfinite Recursion" than it is "Transfinite Induction." In other words, given a statement $A_n$ and a set $z$ you can prove that there is a function $f$ on the ordinals where $f(\beta)$ only depends on how $f$ works on ordinals less than $\beta$.
The two versions (what I'm calling Weaver's and Cohen's) are equally powerful since you can get each from the other.