The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

Abstract

We continue the investigation started in [2] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequencemodels of size at most ℵ 1 in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter D on ω, if the ultraproducts D M 1 n and D M 2 n are elementarily equivalent, then they are isomorphic.Recently the authors of this paper [2] have shown that Keisler's theorem is indeed equivalent to the CH, by showing that if CH fails, then there are two dense linear orders M, N of size ≤ 2 ℵ 0 which do not have isomorphic ultrapowers with respect to any ultrafilter on ω. Much earlier but after Keisler, Shelah [5] removed CH from Keisler's theorem by weakening the conclusion and showed that if L is a countable language and M, N are 2010 Mathematics Subject Classification. 03C20, 03E35.