On principles between Σ₁ - And Σ₂ -induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between Σ₁-induction and Σ₂-induction, are equivalent to the well-foundedness of ω^ω. Among these principles are the iteration of partial functions (PΣ₁) of Hájek and Paris, the bounded monotone enumerations principle (non-iterated, BME₁) 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 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 ω-tower ω^ω.