On principles between $Σ_1$- and $Σ_2$-induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between $Σ_1$-induction and $Σ_2$-induction, are equivalent to the well-foundedness of $ω^ω$. Among these principles are the iteration of partial functions ($PΣ_1$) of Hájek 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-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.

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