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March 2008 Closed maximality principles: implications, separations and combinations
Gunter Fuchs
J. Symbolic Logic 73(1): 276-308 (March 2008). DOI: 10.2178/jsl/1208358754


I investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ,<λ). These principles come in many variants, depending on the parameters which are allowed. I shall write MPΓ(A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are:

• The principles have many consequences, such as <κ-closed-generic Σ12(Hκ) absoluteness, and imply, e.g., that ♢κ holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.

• The principles can be separated into a hierarchy which is strict, for many κ.

• Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously.

The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(Hκ+) holds at all regular κ below any fixed α, the “global” maximality principle, stating that MP<κ-closed(Hκ+ ∪ {κ}) holds at every regular κ, is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from Hκ+ that’s provably <κ-closed-forceably necessary is true, for all regular κ.


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Gunter Fuchs. "Closed maximality principles: implications, separations and combinations." J. Symbolic Logic 73 (1) 276 - 308, March 2008.


Published: March 2008
First available in Project Euclid: 16 April 2008

zbMATH: 1157.03027
MathSciNet: MR2387944
Digital Object Identifier: 10.2178/jsl/1208358754

Primary: 03E35 , 03E40 , 03E45 , 03E55

Keywords: Forcing axioms , Maximality Principles

Rights: Copyright © 2008 Association for Symbolic Logic


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Vol.73 • No. 1 • March 2008
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