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Strong covering without squares

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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 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).
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Twórcy
  • Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, shelah@math.huji.ac.il
  • Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, U.S.A.
Bibliografia
  • [BuMa] M. Burke and M. Magidor, Shelah's pcf theory and its applications, Ann. Pure Appl. Logic 50 (1990) 207-254.
  • [Ca] T. Carlson, unpublished.
  • [DeJe] K. J. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, in: Proc. ISILC Logic Conference (Kiel, 1974), Springer, Berlin, 1975, 115-142.
  • [Sh:b] S. Shelah, Proper Forcing, Springer, Berlin, 1982.
  • [Sh:g] S. Shelah, Cardinal Arithmetic, Clarendon Press, Oxford, 1994.
  • [Sh410] S. Shelah, More on cardinal arithmetic, Arch. Math. Logic 32 (1993) 399-428.
  • [Sh420] S. Shelah, Advances in cardinal arithmetic, in: Finite and Infinite Combinatorics in Sets and Logic, Kluwer, New York, 1993, 355-383.
  • [Sh598] S. Shelah, More on covering lemma, in preparation.
  • [Sh:E12] S. Shelah, Analytic guide, math.LO/9906022.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv166i1p87bwm
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