Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see Buchholz 2003, we introduce systems of fundamental sequences for two kinds of relativized $\vartheta$-function-based notation systems of strength $Π^1_1{\operatorname{-CA}_0}$ and prove Bachmann property for these systems, which is essential for monotonicity properties of subrecursive hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of localization, which was introduced in Wilken 2007. The first kind of stepwise defined $\vartheta$-functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in Wilken 2007, whereas the second kind of $\vartheta$-functions is defined simultaneously and will allow for further generalization to larger proof-theoretic ordinals, see Weiermann and Wilken 2011. The systems of fundamental sequences given here enable 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 2001 and further analyzed in Wilken 2006, 2007, and Carlson and Wilken 2012. Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann 1994. The results of this paper will be applied to the theory of Goodstein sequences, extending results of Fernàndez-Duque and Weiermann 2024.

Abstract A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5].

Abstract In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.