Displaying similar documents to “The covering property for σ-ideals of compact, sets”

Keeping the covering number of the null ideal small

Teruyuki Yorioka (2015)

Fundamenta Mathematicae

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It is proved that ideal-based forcings with the side condition method of Todorcevic (1984) add no random reals. By applying Judah-Repický's preservation theorem, it is consistent with the covering number of the null ideal being ℵ₁ that there are no S-spaces, every poset of uniform density ℵ₁ adds ℵ₁ Cohen reals, there are only five cofinal types of directed posets of size ℵ₁, and so on. This extends the previous work of Zapletal (2004).

Strong covering without squares

Saharon Shelah (2000)

Fundamenta Mathematicae

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Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is...