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Archive for Mathematical Logic

, Volume 56, Issue 1–2, pp 131–154 | Cite as

\(I_0\) and combinatorics at \(\lambda ^+\)

  • Xianghui ShiEmail author
  • Nam Trang
Article

Abstract

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).

Keywords

Axiom \(I_0\) \(\lambda ^+\)-Aronszajn tree Square Weak square Stationary reflection Good scales Diamond \(\lambda \)-Continuum hypothsis Generic absoluteness \(\lambda \)-Goodness 

Mathematics Subject Classification

03E55 03E35 03E05 

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Key Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of California, IrvineIrvineUSA

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