Local reflection, definable elements and 1-provability

Abstract

In this note we study several topics related to the schema of local reflection $\mathsf{Rfn}(T)$ and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with ${\Sigma }_n$-definable parameters, establish its relationship with the relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema we give a new model-theoretic proof of the ${\Sigma }_{n+2}$-conservativity of uniform ${\Sigma }_{n+1}$-reflection over relativized local ${\Sigma }_{n+1}$-reflection. We also study the proof-theoretic strength of Feferman's theorem, i.e., the assertion of $1$-provability in $S$ of the local reflection schema $\mathsf{Rfn}(S)$, and its generalized versions. We relate this assertion to the uniform ${\Sigma }_2$-reflection schema and, in particular, obtain an alternative axiomatization of $\mathsf{I}{\Sigma }_1$.