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Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 427–442 | Cite as

Diamond, scales and GCH down to \(\aleph _{\omega ^2}\)

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Abstract

Gitik and Rinot (Trans Am Math Soc 364(4):1771–1795, 2012) proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal \(\kappa \) of countable cofinality such that \(2^\kappa =\kappa ^+\), there is a very good scale at \(\kappa \), and \(\diamond \) fails along some reflecting stationary subset of \(\kappa ^+\cap \text {cof}(\omega )\). In this paper, we force over Gitik and Rinot’s model but with a modification of Gitik–Sharon (Proc Am Math Soc 136(1):311, 2008) diagonal Prikry forcing to get this result for \(\kappa =\aleph _{\omega ^2}\).

Keywords

Diamond Scales Stationary Reflection Supercompact 

Mathematics Subject Classification

03E05 03E35 03E55 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Illinois at ChicagoChicagoUSA

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