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We show that many principles of first-order arithmetic, previously only knownto lie strictly between $\Sigma_1$-induction and $\Sigma_2$-induction, areequivalent 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 relativizedParis-Harrington principle for pairs, and the totality of the relativizedAckermann-P\'eter function.  With this we show that the well-foundedness of $\omega^\omega$ is a far morewidespread than usually suspected.  Further, we investigate the $k$-iterated version of the bounded monotoneiterations principle (BME$_k$), and show that it is equivalent to thewell-foundedness of the $k+1$-height $\omega$-tower.

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