Iterated Local Reeection Vs Iterated Consistency

For \natural enough" systems of ordinal notation we show that times iterated local reeection schema over a suuciently strong arithmetic T proves the same 0 1-sentences as ! times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at-numbers. We also derive the following more general \mixed" formulas estimating the consistency strength of iterated local reeection: for all ordinals 1 and all , (T) 0 1 T ! (1+) ; (T) 0 1 T +! : Here T stands for times iterated local reeection over T , T stands for times iterated consistency, and 0 1 denotes (provable in T) mutual 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.