Which pairs of quotients over ideals on $\mathbb{N}$ can be
distinguished without assuming additional set theoretic
axioms? Essentially, those that are not isomorphic under the
Continuum Hypothesis. A CH-diagonalization method for
constructing isomorphisms between certain quotients of
countable products of finite structures is developed and used
to classify quotients over ideals in a class of generalized
density ideals. It is also proved that many analytic ideals
give rise to quotients that are countably saturated (and
therefore isomorphic under CH).