Iterated local reflection versus iterated consistency

Lev Beklemishev
1995 Annals of Pure and Applied Logic  
For "natural enough" systems of ordinal notation we show that tl times iterated local reflection schema over a suflicientiy strong arithmetic T proves the same KIF-sentences as CO" times iterated consistency. A corollary is that the two hierarchies catch up module relative interpretability exactly at E-numbers. We also derive the following more general '"mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals CI 2 1 and all p, (T")fl En: TU*.~l+~),
more » ... -t-i: q+o'. Here T" stands for c( times iterated local reflection over T, T8 stands for a times iterated consistency, and -np denotes (provable in T) mutual ny-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. * Email: lev(~~bek1.mian.s~. 0168-0072/95/$09.50 0 199% Elsevier Science B.V. All rights reserved SSDl 0148-0072(9S)OOOO7-0 26 L. Beklemishev/Annals of Pure and Applied Logic 75 (1995) 25-48
doi:10.1016/0168-0072(95)00007-4 fatcat:utzrvcl5pfcl7mcq3jw5sotqtu