Weak square and stationary reflection



It is well-known that the square principle \({\square_\lambda}\) entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle \({\square^{*} _\lambda}\) does not. Here we show that if μcf(λ) < λ for all μ < λ, then \({\square^{*} _\lambda}\) entails the existence of a non-reflecting stationary subset of \({E^{\lambda^+}_{{\rm cf}(\lambda)}}\) in the forcing extension for adding a single Cohen subset of λ+.

It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of \({\square^{*} _\lambda}\) for every singular cardinal λ of countable cofinality.

Key words and phrases

weak square simultaneous stationary reflection SCFA 

Mathematics Subject Classification

primary 03E35 secondary 03E57 03E05 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.The College of Staten Island (CUNY)Staten IslandU.S.A.
  2. 2.The Graduate Center (CUNY)New YorkU.S.A.
  3. 3.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

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