Paper detail

A Uniform Characterization of $Σ_1$-Reflection over the Fragments of Peano Arithmetic

We show that the theory $IΣ_1$ of $Σ_1$-induction proves the following statement: For all $n\geq 2$, the uniform $Σ_1$-reflection principle over the theory $IΣ_n$ is equivalent to the totality of the function $F_{ω_n}$ at stage $ω_n$ 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 $IΣ_1$. It is well possible that this has been considered before. Our aim is merely to provide a detailed exposition of this important result.