For a moduleMfor various quotient modulesM/NofM, cardinality restrictions are placed on the number of generators of submodules ofM/N, and on the (infinite) Goldie dimensions ofM/N. The conditionsFGSandTCof R. Kurshan are formulated and generalized for arbitrary cardinals. The module theoretic consequences of these and other such finite-ness conditions with ascending chain conditions are explored.