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 second order arithmetic. The main result is that any arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, $n\geq k+1$ , $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, where T augmented with full induction, $\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$ denotes schema transfinite induction up to $\varepsilon _0$ $\varPi ^1_n$ without set parameters.