Iterated local reflection vs iterated consistency
Lev D. Beklemishev · 1994
For "natural enough" systems of ordinal notation we show that ff times iterated local reflection schema over a sufficiently strong arithmetic T proves the same \\Pi 0 1 -sentences as ! ff times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ffl-numbers. We also derive the following more general "mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals ff 1 and all fi, (T ff ) fi j \\Pi 0 1 T ! ff \\Delta(1+fi) ; (T fi ) ff j \\Pi 0 1 T fi+! ff : Here T ff stands for ff times iterated local reflection over T , T fi stands for fi times iterated consistency, and j \\Pi 0 1 denotes (provable in T ) mutual \\Pi 0 1 -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. 1 Introduction Since the fundamental works of A.Turing [17] and S.Feferman [6...