A Uniform Characterization of ${\mathrm{\Sigma }}_1$-Reflection over the Fragments of Peano Arithmetic

We show that the theory $I{\mathrm{\Sigma }}_1$ of ${\mathrm{\Sigma }}_1$-induction proves the following statement: For all $n\ge 2$, the uniform ${\mathrm{\Sigma }}_1$-reflection principle over the theory $I{\mathrm{\Sigma }}_n$ 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 $I{\mathrm{\Sigma }}_1$. It is well possible that this has been considered before. Our aim is merely to provide a detailed exposition of this important result.