The Consistency Strength of $$\aleph_{\omega}$$ and $$\aleph_{{\omega}_1}$$ Being Rowbottom Cardinals Without the Axiom of Choice

Abstract

We show that for all natural numbers n, the theory “ZF + DC $$_{\aleph_n}$$ + $$\aleph_{\omega}$$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $$\aleph_{\omega_1}$$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We also discuss some generalizations of these results.