  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.

Logic