How many Boolean algebras ${\scr P}({\Bbb N})/\scr I$ are there?

Abstract

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).