On strong chains of sets and functions

TL;DR

This paper extends Shelah's result by proving the non-existence of long chains in the set-theoretic structure of $[\omega_2]^{\aleph_2}$, contrasting with known results about functions and smaller cardinals.

Contribution

It improves Shelah's theorem from functions to sets, showing no chains of length $\omega_3$ in $[\omega_2]^{\aleph_2}$ increasing modulo finite.

Findings

No chains of length $\omega_3$ in $[\omega_2]^{\aleph_2}$ increasing modulo finite.

Contrasts with existing results on chains in smaller cardinal structures.

Studies the depth of function spaces ${}^\kappa\mu$ modulo certain ideals.

Abstract

Shelah has shown that there are no chains of length $\omega_3$ 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 $\omega_3$ in $[\omega_2]^{\aleph_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length $\omega_2$ increasing modulo finite in $[\omega_1]^{\aleph_1}$ as well as in ${}^{\omega_1}\omega_1$. More generally, we study the depth of function spaces ${}^\kappa\mu$ quotiented by the ideal $[\kappa]^{< \theta}$ where $\theta< \kappa$ are infinite cardinals.