A note on fragments of uniform reflection in second order arithmetic

TL;DR

This paper explores fragments of uniform reflection principles in second order arithmetic, establishing equivalences with transfinite induction schemas for certain formulas, thereby deepening understanding of the logical strength of these theories.

Contribution

It provides new characterizations of uniform reflection fragments as transfinite induction schemas in second order arithmetic, extending previous results to more complex formulas.

Findings

Equivalence between reflection principles and transfinite induction schemas.

Extension of results to theories with full induction.

Clarification of the logical strength of reflection fragments.

Abstract

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending ${\sf RCA}_0$ and axiomatizable by a $\Pi^1_{k+2}$ sentence, and for any $n\geq k+1$, \[ T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \] \[ T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \] where $T$ is $T_0$ augmented with full induction, and $\mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}$ denotes the schema of transfinite induction up to $\varepsilon_0$ for $\varPi^1_n$ formulas without set parameters.