If we take Peano Arithmetic and restrict induction to formulas over various fragments of the arithmetic hierarchy, say to the $\Sigma^0\_n$ formulas for various $n$ or some other interesting fragments, how does the proof theoretic ordinal for the theory vary?



Ordinal Analysis of Peano Arithmetic with Restricted Induction
The proof-theoretic ordinal of $I\Sigma^0\_n$ (for $n > 0$) is well-known to be $\omega\_{n+1}$, where $\omega\_1:=\omega$, $\omega\_{n+1}:=\omega^{\omega\_n}$. See e.g. [Avigad & Sommer](http://www.jstor.org/stable/421195). 


