As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order arithmetic with the ramified hierarchy going to arbitrarily high transfinite ordinals, then the sets you'll ultimately get are the hyperarithmetic sets. In contrast, in the standard Feferman-Schutte version of predicative second-order arithmetic, we only allow the ramified hierarchy to go up to an ordinal $\alpha$ if we are able to prove that $\alpha$ is well-founded using the comprehension schemata for lower levels of the ramified hierarchy. And proceeding in this way, Feferman and Schutte showed you could build the ramified hierarchy up to $\Gamma_0$, the Feferman-Schutte ordinal. And Gamma_0 is also the proof-theoretic ordinal of the resultant system of predicative second-order arithmetic.
Now obviously going up all the way up the ramified hierarchy would yield more sets than simply going up to $\Gamma_0$, so I expect second-order arithmetic with hyperarithmetical comprehension to be stronger than then the Feferman-Schutte version of predicative second-order arithmetic, and thus I expect that the former would have a higher proof-theoretic ordinal than $\Gamma_0$. So what is this ordinal?
One thing that gives me pause is that page 3 of this PDF (which I haven't read) says that "hyperarithmetic analysis", which I assume means second-order arithmetic with hyperarithmetical comprehension, is "mostly here in between" $ACA_0$ and $ATR_0$, which have proof-theoretic ordinals $\epsilon_0$ and $\Gamma_0$ respectively. How is this possible? Shouldn't hyperarithmetical comprehension have a higher proof-theoretic ordinal than $\Gamma_0$?
Any help would be greatly appreciated.
Thank You in Advance.