On strong chains of sets and functions

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