Journal of Symbolic Logic

Co-stationarity of the ground model

Natasha Dobrinen and Sy-David Friedman

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This paper investigates when it is possible for a partial ordering ℛ to force 𝒫κ(λ)∖ V to be stationary in V. It follows from a result of Gitik that whenever ℛ adds a new real, then 𝒫κ(λ)∖ V is stationary in V for each regular uncountable cardinal κ in V and all cardinals λ>κ in V [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 V, 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 V. The following is equiconsistent with κ measurable cardinals: If ℛ is κ-Cohen forcing, then 𝒫κ⁺(ℵκ)∖ V is stationary in V.

Article information

J. Symbolic Logic, Volume 71, Issue 3 (2006), 1029-1043.

First available in Project Euclid: 4 August 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results 03E45: Inner models, including constructibility, ordinal definability, and core models

𝒫_κ λ co-stationarity Erdös cardinal measurable cardinal


Dobrinen, Natasha; Friedman, Sy-David. Co-stationarity of the ground model. J. Symbolic Logic 71 (2006), no. 3, 1029--1043. doi:10.2178/jsl/1154698589.

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