2023/04/04 by Lambie-Hanson, Chris
#03E04 #03E05 #03E35 #03E55 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2304.02132
Motivated by two open questions about two-cardinal tree properties, we introduce and study generalized narrow system properties. The first of these questions asks whether the strong tree property at a regular cardinal κ≥ ω2 implies the Singular Cardinals Hypothesis (SCH) above κ. We show here that a certain narrow system property at κ that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply SCH above κ. The second of these questions asks whether the strong tree property can consistenty hold simultaneously at all regular cardinals κ≥ ω2. We show here that the analogous question about the generalized narrow system property has a positive answer. We also highlight some connections between generalized narrow system properties and the existence of certain strongly unbounded subadditive colorings.