Iterated local reflection versus iterated consistency

Abstract

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 ω^α 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, (T^α)_β ≡ Π₁⁰ T_{ω^α·(1 + β)}, (T_β)^α ≡ Π₁⁰ T_{β + ω^α}. Here T^α stands for α times iterated local reflection over T, T_β 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.