Source: J. Symbolic Logic
Volume 74, Issue 3
The motivation for this paper is the following: In
 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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