Bachmann–Howard derivatives

Abstract It is generally accepted that H. Friedman’s gap condition closely related to iterated collapsing functions from ordinal analysis. But what precisely the connection? We offer following answer: In a previous paper we have shown arises an iterative construction on transformations of partial orders. Here show parallel for linear orders yields familiar functions. The iteration step in case instance general call ‘Bachmann–Howard derivative’. present paper, focus unary case, i.e., sequences rather than trees and, correspondingly, addition-free notation systems. This partly convenience, but it also allows us clarify phenomenon specific setting: As by van der Meeren, Rathjen and Weiermann, admits two linearizations with different properties. will see these correspond recursive constructions sequences.