Comparing WO$(\omega^\omega)$ with $\Sigma^0_2$ induction

Abstract

Let WO$(\omega^\omega)$ be the statement that the ordinal number $\omega^\omega$ is well ordered. WO$(\omega^\omega)$ has occurred several times in the reverse-mathematical literature. The purpose of this expository note is to discuss the place of WO$(\omega^\omega)$ within the standard hierarchy of subsystems of second-order arithmetic. We prove that WO$(\omega^\omega)$ is implied by I$\Sigma^0_2$ and independent of B$\Sigma^0_2$. We also prove that WO$(\omega^\omega)$ and B$\Sigma^0_2$ together do not imply I$\Sigma^0_2$.