Between reduced powers and ultrapowers, II

Abstract

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.The trivializing effect of the Continuum Hypothesis (CH) to the structure of the continuum has been known at least since the times of Sierpiński and Gödel ([19]). The particular instance of this phenomenon that we are concerned with in the present paper is the existence of highly non-canonical isomorphisms between massive quotient structures of cardinality c = 2 ℵ 0 . The operation of taking a reduced product F A n of a sequence (A n ) of first-order structures often results in a countably saturated structure. 1 This is the case with the two most commonly used reduced products: ultraproducts associated with nonprincipal ultrafilters on N and reduced products associated with the Fréchet filter. If each A n has the cardinality of at most c, then so does F A n , and the CH implies that the latter structure is saturated. By a classical theorem of Keisler, elementarily equivalent and saturated first-order structures of the same cardinality are isomorphic (see [5, Theorem 5.1.13]). Therefore the isomorphism of such reduced products reduces (no pun intended) to elementary equivalence.In [11], this observation was combined with computation of the theory of the structure (K denotes the Cantor space)