In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system$OT$is introduced based on$\psi $-functions. Provable$\Sigma _{1}$-sentences on$L_{\omega _{1}^{CK}}$are bounded through cut-elimination on operator controlled derivations.

In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal \(\sup _{N is an upper bound for the proof-theoretic ordinal of a set theory \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)\). In this paper we show that a second order arithmetic \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) proves the wellfoundedness up to \(\psi _{\varOmega _{1}}(\varepsilon _{\varOmega _{{\mathbb {S}}+N+1}})\) for each _N_. It is easy to interpret \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) in \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)\).