# Strong covering without squares

Fundamenta Mathematicae (2000)

- Volume: 166, Issue: 1-2, page 87-107
- ISSN: 0016-2736

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topShelah, Saharon. "Strong covering without squares." Fundamenta Mathematicae 166.1-2 (2000): 87-107. <http://eudml.org/doc/212477>.

@article{Shelah2000,

abstract = {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 V-regular, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).},

author = {Shelah, Saharon},

journal = {Fundamenta Mathematicae},

keywords = {set theory; covering; strong covering lemma; pcf theory; inner model},

language = {eng},

number = {1-2},

pages = {87-107},

title = {Strong covering without squares},

url = {http://eudml.org/doc/212477},

volume = {166},

year = {2000},

}

TY - JOUR

AU - Shelah, Saharon

TI - Strong covering without squares

JO - Fundamenta Mathematicae

PY - 2000

VL - 166

IS - 1-2

SP - 87

EP - 107

AB - 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 V-regular, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

LA - eng

KW - set theory; covering; strong covering lemma; pcf theory; inner model

UR - http://eudml.org/doc/212477

ER -

## References

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