Fundamental Sequences Based on Localization

We introduce systems of fundamental sequences for relativized \(\vartheta \) -function-based notation systems of strength \(\Pi ^1_1{{\text {-CA}}_0}\) and prove the Bachmann property for these systems, which is essential for monotonicity properties of fast growing hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of localization, which was introduced in [9]. Our assignment extends Buchholz’ assignment for ordinals below Bachmann-Howard ordinal, see [1]. The specific system of \(\vartheta \) -functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in [9] and enables the investigation of fundamental sequences and independence phenomena also in the context of patterns of resemblance, an approach to ordinal notations that is both semantic and combinatorial and was first introduced by Carlson in [3] and further analyzed in [4, 8, 10, 11]. Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann in [2]. An extended version of this article containing complete proofs will be published elsewhere.