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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:
 3e+40 Other aspects of forcing and Boolean-valued models 3e+55 Large cardinals
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