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

TL;DR

This paper investigates the logical strength of the statement WO$(oldsymbol{ ext{ extomega}}^ ext{ extomega})$, showing its relation to various subsystems of second-order arithmetic, particularly its implications and independence from $oldsymbol{ ext{ extSigma}}^0_2$ induction and bounding schemes.

Contribution

It establishes that WO$( ext{ extomega}^ ext{ extomega})$ is implied by I$oldsymbol{ ext{ extSigma}}^0_2$ and independent of B$oldsymbol{ ext{ extSigma}}^0_2$, clarifying its position in the reverse-mathematical hierarchy.

Findings

WO$( ext{ extomega}^ ext{ extomega})$ is implied by I$ ext{ extSigma}^0_2$

WO$( ext{ extomega}^ ext{ extomega})$ is independent of B$ ext{ extSigma}^0_2$

WO$( ext{ extomega}^ ext{ extomega})$ and B$ ext{ extSigma}^0_2$ do not imply I$ ext{ extSigma}^0_2$

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$\Sigma^0_2$.