Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

TL;DR

This paper establishes regularity properties of Buchholz's fundamental sequences for the ta function, extending to variants, and shows that Hardy functions based on these systems dominate all primitive recursive functions up to a certain large countable ordinal.

Contribution

It proves regularity conditions for Buchholz's fundamental sequences and extends these results to variants of the ta function, demonstrating their significance in ordinal analysis.

Findings

Fundamental sequences satisfy the Bachmann property.

Hardy functions based on these systems dominate primitive recursive functions.

Extensions to variants of the ta function without addition.

Abstract

We prove that Buchholz's system of fundamental sequences for the $\vartheta$ function enjoys various regularity conditions, including the Bachmann property. We partially extend these results to variants of the $\vartheta$ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along $\vartheta(\varepsilon_{\Omega+1})$.