Baire Categoricity and Σ10-Induction

Abstract

We investigate the reverse-mathematical status of a version of the Baire category theorem known as $\mathrm{BCT}$. In a 1993 paper Brown and Simpson showed that $\mathrm{BCT}$ is provable in ${\mathsf{RCA}}_0$. We now show that $\mathrm{BCT}$ is equivalent to ${\mathsf{RCA}}_0$ over ${\mathsf{RCA}}_0^{\ast }$.