On strong chains of sets and functions

Abstract

Shelah has shown that there are no chains of length ω3 increasing modulo finite in ω2ω2${}^{\omega _2}\omega _2$. We improve this result to sets. That is, we show that there are no chains of length ω3 in [ω2]ℵ2$[\omega _2]^{\aleph _2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in [ω1]ℵ1$[\omega _1]^{\aleph _1}$ as well as in ω1ω1${}^{\omega _1}\omega _1$. More generally, we study the depth of function spaces κμ${}^\kappa \mu$ quotiented by the ideal [κ]<θ$[\kappa ]^{&lt; \theta }$ where θ<κ$\theta &lt; \kappa$ are infinite cardinals.