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

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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 ϑ(εΩ+1).

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Versió postprint del document publicat a:

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Annals of Pure and Applied Logic, 2024, vol. 175, num.8

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Articles publicats en revistes (Filosofia)

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