Let $PA_{\omega}$ be just like $PA$ except that $PA_{\omega}$-proofs can use any number of applications of the recursive $\omega$-rule.
The recursive $\omega$-rule allows the following:
For each formula $\varphi$ with one free variable, if there is a unary total recursive function $f$ such that $\forall n \in N$, $f(n)$ is the code of a proof of $\varphi(\overline{n})$, then we may conclude $\forall x \ \varphi(x)$.
Each sentence of true arithmetic has a recursive $PA_{\omega}$-proof of height $<\omega^{2}$. Saying that the proof is recursive means that it can be coded by a total recursive function.
Assuming a canonical encoding of ordinals,
Q1: How much transfinite induction must be added to $PA$ so that for each sentence $\varphi$ of true arithmetic, the resulting system proves something equivalent to "there is a recursive $PA_{\omega}$-proof of $\varphi$"?
Q2: How much transfinite induction must be added to $PA$ so that for each sentence $\varphi$ of true arithmetic, the resulting system proves something equivalent to "there is a recursive $PA_{\omega}$-proof of $\varphi$ whose height is $< \omega^{2}$ "?
Addendum: Since for each $\varphi$ of true arithmetic, the statement that
there is a $PA_{\omega}$-proof of $\varphi$ whose height is $< \omega^{2}$
is true, one might think that each of those statements is equivalent to the statement that 0=0, and that the answer to both Q2 and Q1 would be that no additional induction must be added to $PA$.
I intend Q1 and Q2 to be using a stronger notion of “equivalent”, so that answers like this are ruled out.
