Strong covering without squares

Shelah, Saharon

Shelah, Saharon

Fundamenta Mathematicae, Tome 163 (2000), p. 87-107 / Harvested from The Polish Digital Mathematics Library

Résumé

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).

Publié le : 2000-01-01

Zbl 0959.03028

EUDML-ID : urn:eudml:doc:212477

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     author = {Saharon Shelah},
     title = {Strong covering without squares},
     journal = {Fundamenta Mathematicae},
     volume = {163},
     year = {2000},
     pages = {87-107},
     zbl = {0959.03028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv166i1p87bwm}
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Shelah, Saharon. Strong covering without squares. Fundamenta Mathematicae, Tome 163 (2000) pp. 87-107. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv166i1p87bwm/

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