On principles between ∑1- and ∑2-induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between ∑_1-induction and ∑_2-induction, are equivalent to the well-foundedness of ω^ω. Among these principles are the iteration of partial functions (P∑_1) of Hajek and Paris, the bounded monotone enumerations principle (non-iterated, BME_1) by Chong, Slaman, and Yang, the relativized Paris-Harrington principle for pairs, and the totality of the relativized Ackermann-Peter function. With this we show that the well-foundedness of ω^ω is a far more widespread than usually suspected. Further, we investigate the k-iterated version of the bounded monotone iterations principle (BME_k), and show that it is equivalent to the well-foundedness of the k + 1-height -tower ω⋰ω