The measuring principle and the continuum hypothesis

Abstract

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