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Hierarchies of forcing axioms. I. (English) Zbl 1154.03032
The authors obtain consistency bounds for versions of the Semi-Proper and Proper Forcing Axioms respectively: the versions are for \(\mathfrak{c}\)-linked, \(\mathfrak{c}^+\)-linked and \(\mathfrak{c}^{++}\)-linked partial orders. Both PFA and SPFA for \(\mathfrak{c}\)-linked partial orders are equiconsistent with the existence of a \(\Sigma^2_1\)-indescribable cardinal. The authors prove a general result, from which the results alluded to above follow: if \(\lambda\leq\theta\), \(\theta^{<\lambda}=\theta\) and \(2^\theta=\theta^+\) then, provided \(\lambda\) is \((\theta,\Sigma_1^2)\)-subcompact, one can create a generic extension in which \(\lambda=\mathfrak{c}=\aleph_2\) and SPFA for \(\theta\)-linked partial orders holds.
Reviewer: K. P. Hart (Delft)

MSC:
03E40 Other aspects of forcing and Boolean-valued models
03E55 Large cardinals
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