Singular cardinals and square properties

Menachem Magidor, Dima Sinapova
2017 Proceedings of the American Mathematical Society  
We analyze the effect of singularizing cardinals on square properties. In two old papers of Dzamonja-Shelah and of Gitik it was implicit that if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we show that if V ⊂ W , κ < ν are cardinals, where ν is regular in V , κ is
more » ... egular in V , κ is a singular cardinal in W of countable cofnality, and cf W (τ ) = ω for all V -regular κ ≤ τ ≤ ν, and (ν + ) V = (κ + ) W , then W |= κ,ω .
doi:10.1090/proc/13650 fatcat:uc3vcqcfo5ce5nnczk6gjzvvcy