Combined Maximality Principles up to large cardinals

Journal of Symbolic Logic 74 (3):1015-1046 (2009)
The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for < κ -closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so. how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.
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DOI 10.2178/jsl/1245158097
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