# Sh:580

- Shelah, S. (2000).
*Strong covering without squares*. Fund. Math.,**166**(1-2), 87–107. arXiv: math/9604243 MR: 1804706 -
Abstract:

We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let \kappa be a cardinal in V. We say that \kappa-covering holds between V and W iff for all X\in V with X\subseteq ON and V\models|X|<\kappa, there exists Y\in W such that X\subseteq Y\subseteq ON and V\models |Y|<\kappa. Strong \kappa-covering holds between V and W iff for every structure {\mathcal M} \in V for some countable first-order language whose underlying set is some ordinal \lambda, and every X \in V with X\subseteq\lambda and V\models |X|<\kappa, there is Y\in W such that X\subseteq Y \prec M and V\models |Y|<\kappa.We prove that if \kappa is V-regular, \kappa^+_V=\kappa^+_W, and we have both \kappa-covering and \kappa^+-covering between W and V, then strong \kappa-covering holds. Next we show that we can drop the assumption of \kappa^+-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 \kappa^+_W =\kappa^+_V and weaken the \kappa^+-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

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Bib entry

@article{Sh:580, author = {Shelah, Saharon}, title = {{Strong covering without squares}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {166}, number = {1-2}, year = {2000}, pages = {87--107}, issn = {0016-2736}, mrclass = {03E04 (03E05 03E45)}, mrnumber = {1804706}, note = {\href{https://arxiv.org/abs/math/9604243}{arXiv: math/9604243}}, arxiv_number = {math/9604243}, specialissue = {Saharon Shelah's anniversary issue} }