Strong covering without squares

Saharon Shelah¹

¹

Fundamenta Mathematicae, Tome 166 (2000) no. 1, pp. 87-107.

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).

Détail

DOI : 10.4064/fm-166-1-2-87-107

Keywords: set theory, covering, strong covering lemma, pcf theory