We show that one can force the Measuring principle without adding any new reals. We also show that it is consistent with the large continuum. These results answer two famous questions of Justin Moore. § 0. IntroductionIn this paper we study Moore's measuring principle and prove two consistency results related to it. We show that one can force it by a proper forcing notion which adds no new reals. We also show that it is consistent with the continuum being arbitrary large. These results show that the measuring principle has no effects on the size of the continuum.Before we continue, let us start by recalling the definition of the measuring principle. Definition 0.1. Measuring holds iff for every sequence C = C δ : δ < ω 1 , δ limit , if each C δ is a club of δ, then there is a club C ⊆ ω 1 which measures C, i.e., for every δ ∈ C, there is some α < δ such that either
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.
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