\nabla_κ, remarkable cardinals, and 0^#
Abstract
Description
For an uncountable regular cardinal κwe let \nabla_κ(A) be the statement that A \subset κand for all regular θ> κ, the set of all X \in [θ]^<κsuch that X \cap κ\in κand otp(X \cap OR) is a cardinal in L[A \cap X \cap κ] is stationary. We had shown earlier that \nabla_{ω_1}(A) can hold in a generic extension of L. We now prove that \nabla_{ω_2}(A) can hold in a semi-proper generic extension of L, whereas \nabla_{ω_3}(0) is equivalent with the existence of 0^#.
7 pages
7 pages