Fundamental sequences based on localization

TL;DR

This paper develops new fundamental sequences for advanced ordinal notation systems based on $ heta$-functions, proving the Bachmann property and enabling analysis of independence phenomena and Goodstein sequences.

Contribution

It introduces systems of fundamental sequences for $ heta$-function-based ordinal notations with the Bachmann property, extending the framework for larger proof-theoretic ordinals.

Findings

Proved Bachmann property for the new systems

Enabled investigation of independence phenomena in ordinal patterns

Extended analysis of Goodstein sequences

Abstract

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 $\Pi^1_1{\operatorname{-CA}_0}$ 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 Fern\`andez-Duque and Weiermann 2024.