# Sh:580

• Shelah, S. (2000). Strong covering without squares. Fund. Math., 166(1-2), 87–107.
• 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).

• Version 1996-04-25_10 (23p) published version (21p)
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},
mrnumber = {1804706},
mrclass = {03E04 (03E05 03E45)},
note = {\href{https://arxiv.org/abs/math/9604243}{arXiv: math/9604243}},
arxiv_number = {math/9604243},
specialissue = {Saharon Shelah's anniversary issue}
}