Co-stationarity of the ground model

Abstract

This paper investigates when it is possible for a partial ordering ℛ to force 𝒫_κ(λ)∖ V to be stationary in V^ℛ. It follows from a result of Gitik that whenever ℛ adds a new real, then 𝒫_κ(λ)∖ V is stationary in V^ℛ for each regular uncountable cardinal κ in V^ℛ and all cardinals λ>κ in V^ℛ [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω₁-Erdős cardinals: If ℛ is ℵ₁-Cohen forcing, then 𝒫_κ(λ)∖ V is stationary in V^ℛ, for all regular κ≥ℵ₂ and all λ>κ. The following is equiconsistent with an ω₁-Erdős cardinal: If ℛ is ℵ₁-Cohen forcing, then 𝒫_ℵ₂(ℵ₃)∖ V is stationary in V^ℛ. The following is equiconsistent with κ measurable cardinals: If ℛ is κ-Cohen forcing, then 𝒫_κ⁺(ℵ_κ)∖ V is stationary in V^ℛ.