Abstract
This paper investigates when it is possible for a partial ordering ℛ to force 𝒫κ(λ)∖ V to be stationary in Vℛ. It follows from a result of Gitik that whenever ℛ adds a new real, then 𝒫κ(λ)∖ V is stationary in Vℛ for each regular uncountable cardinal κ in Vℛ and all cardinals λ>κ in Vℛ [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 𝒫κ(λ)∖ V is stationary in Vℛ, 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 𝒫ℵ₂(ℵ₃)∖ V is stationary in Vℛ. The following is equiconsistent with κ measurable cardinals: If ℛ is κ-Cohen forcing, then 𝒫κ⁺(ℵκ)∖ V is stationary in Vℛ.
Citation
Natasha Dobrinen. Sy-David Friedman. "Co-stationarity of the ground model." J. Symbolic Logic 71 (3) 1029 - 1043, September 2006. https://doi.org/10.2178/jsl/1154698589
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