Comparing WO$({\omega }^{\omega })$ with ${\Sigma }_2^0$ 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${\Sigma }_2^0$ and independent of B${\Sigma }_2^0$. We also prove that WO$({\omega }^{\omega })$ and B${\Sigma }_2^0$ together do not imply I${\Sigma }_2^0$.

arXiv comment6 pages