A Uniform Characterization of $\Sigma_1$-Reflection over the Fragments of Peano Arithmetic

We show that the theory $I\Sigma_1$ of $\Sigma_1$-induction proves the following statement: For all $n\geq 2$, the uniform $\Sigma_1$-reflection principle over the theory $I\Sigma_n$ is equivalent to the totality of the function $F_{\omega_n}$ at stage $\omega_n$ of the fast-growing hierarchy. The method applied is a formalization of infinite proof theory. The literature contains several proofs which place the quantification over $n$ in the meta-theory (and also prove the separate cases $n=0,1$). In contrast, the author knows of no explicit argument that would allow us to internalize the quantification while keeping the meta-theory as low as $I\Sigma_1$. It is well possible that this has been considered before. Our aim is merely to provide a detailed exposition of this important result.