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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)

03E40 Other aspects of forcing and Boolean-valued models
03E55 Large cardinals
Full Text: DOI Euclid
[1] DOI: 10.1090/S0002-9939-1995-1257099-0
[2] DOI: 10.2307/1971415 · Zbl 0645.03028
[3] Surveys in Set Theory 87 pp 1– (1983) · Zbl 0511.00004
[4] The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal 1 (1999) · Zbl 0954.03046
[5] DOI: 10.1016/0001-8708(92)90038-M · Zbl 0785.03031
[6] Jensen’s principles and the Novak number of partially ordered sets 51 pp 47– (1986)
[7] DOI: 10.1090/conm/031/763902
[8] Proper and Improper Forcing (1998) · Zbl 0889.03041
[9] Semiproper forcing axiom implies Martin maximum but not PFA+ 52 pp 360– (1987) · Zbl 0625.03035
[10] DOI: 10.1016/j.aim.2007.05.005 · Zbl 1124.03022
[11] DOI: 10.2307/2687750 · Zbl 0992.03062
[12] Hierarchies of forcing axioms II · Zbl 1154.03031
[13] Multiple forcing 88 (1986) · Zbl 0601.03019
[14] DOI: 10.1016/S0168-0072(99)00010-X · Zbl 0949.03045
[15] Topics in Set Theory: Lebesgue measurability, large cardinals, forcing axioms, rho-functions 1476 (1991) · Zbl 0729.03022
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