A Note on Iterated Consistency and Infinite Proofs

Schmerl and Beklemishev’s work on iterated reflection achieves two aims: It introduces the important notion of ${\mathrm{\Pi }}_1^0$-ordinal, characterizing the ${\mathrm{\Pi }}_1^0$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the ${\mathrm{\Pi }}_1^0$-ordinals for a range of theories. The present note demonstrates that these achievements are independent: We read off ${\mathrm{\Pi }}_1^0$-ordinals from a Sch"utte-style ordinal analysis via infinite proofs, in a direct and transparent way.