A note on fragments of uniform reflection in second order arithmetic

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.