On principles between $\Sigma_1$- and $\Sigma_2$-induction, and monotone enumerations

Abstract

We show that many principles of first-order arithmetic, previously only known to lie strictly between $\Sigma_1$-induction and $\Sigma_2$-induction, are equivalent to the well-foundedness of $\omega^\omega$. Among these principles are the iteration of partial functions ($P\Sigma_1$) of H\'ajek 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-P\'eter function. With this we show that the well-foundedness of $\omega^\omega$ 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 $\omega$-tower.