Goodstein sequences for prominent ordinals up to the Bachmann–Howard ordinal

In his original paper Goodstein (1947) [6] introduces a hierarchy of functions. The third function of that hierarchy, usually referred to as the Goodstein function, is shown to have the same growth rate as hε0, the ε0th function of the Hardy hierarchy. More generally, the question is raised whether there is a clear connection between larger ordinals on the one hand and the so-called level-k Goodstein sequences, for k>3, and their termination on the other. In other words, we try to measure the growth rate of level-k Goodstein functions by relating them to a standard ordinal-indexed hierarchy of functions, in this case the Hardy hierarchy. In this paper, we define appropriate Goodstein sequences for prominent ordinals up to the Bachmann–Howard ordinal.