Abstract

We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text].