Iterated local reflection versus iterated consistency Original Research Article

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π10-sentences as ωα times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, (Tα)β ≡ Π10Tωα·(1 + β), (Tβ)α ≡ Π10Tβ + ωα. Here Tα stands for α times iterated local reflection over T, Tβ stands for β times iterated consistency, and ≡ Π10 denotes (provable in T) mutual Π10-conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal.