It is well-known that the square principle □ λ entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle □λ∗ does not. Here we show that if μcf(λ) < λ for all μ < λ, then □λ∗ entails the existence of a non-reflecting stationary subset of Ecf(λ)λ+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 □λ∗ for every singular cardinal λ of countable cofinality.
Bibliographical noteFunding Information:
∗Corresponding author. †The first author was partially supported by PSC-CUNY grant 69656-00 47. ‡The second author was partially supported by the Israel Science Foundation 1630/14). Key words and phrases: weak square, simultaneous stationary reflection, SCFA. Mathematics Subject Classification: primary 03E35, secondary 03E57, 03E05.
© 2018, Akadémiai Kiadó, Budapest, Hungary.
- simultaneous stationary reflection
- weak square