On strong chains of sets and functions

Abstract

AbstractShelah has shown that there are no chains of length ω3increasing modulo finite in . We improve this result to sets. That is, we show that there are no chains of length ω3in increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2increasing modulo finite in as well as in . More generally, we study the depth of function spaces quotiented by the ideal where are infinite cardinals.