Aristotelian poetry

Abstract

Jing Zhang proved the consistency of $\left(\genfrac{}{}{0ex}{}{{\omega }_2}{{\omega }_1}\right)\to {\left(\genfrac{}{}{0ex}{}{n}{{\omega }_1}\right)}_{\omega }$ for every $n\in \omega$ with the negative relation $\left(\genfrac{}{}{0ex}{}{{\omega }_2}{{\omega }_1}\right)\nrightarrow {\left(\genfrac{}{}{0ex}{}{\omega }{{\omega }_1}\right)}_{\omega }$. We reduce the consistency strength of this statement to an ${\omega }_1$-Erdos cardinal.