Comparing WO$(ω^ω)$ with $Σ^0_2$ induction

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

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