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

Abstract

Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.

In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ 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 $\vartheta \left({\epsilon }_{\Omega +1}\right)$.