On inverse Goodstein sequences

Abstract

In the late 1980s, Abrusci, Girard and van de Wiele defined a variant of Goodstein sequences: the so-called inverse Goodstein sequence. In their work, they show that it terminates precisely at the Bachmann-Howard ordinal. This reveals that a proof of this fact requires substantial consistency strength. Moreover, the authors could show that sequences of this kind terminate even if the hereditary base change at the heart of their construction is replaced by a generalization using arbitrary dilators. It has been a conjecture by Andreas Weiermann that this more general result has a connection to Bachmann-Howard fixed points and is, therefore, equivalent to one of the most famous strong set existence principles from reverse mathematics: $Π^1_1$-comprehension. In this article, we prove this conjecture to be correct. Moreover, we show that the ordinal at which such sequences terminate is, in a fundamental way, isomorphic to the $1$-fixed point of their dilator, a new concept introduced by Freund and Rathjen. This yields explicit notation systems and a general method for specifying such ordinals. Also, using the notation systems provided by $1$-fixed points, we can reproduce the result that the Goodstein sequence terminates at the Bachmann-Howard ordinal in a weak system. Additionally, we perform a similar computation for a variant of Goodstein sequences, which terminates at a predicative ordinal.