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

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