Hierarchies of forcing axioms I

Journal of Symbolic Logic 73 (1):343-362 (2008)
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Abstract

We prove new upper bound theorems on the consistency strengths of SPFA (θ), SPFA(θ-linked) and SPFA(θ⁺-cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(c-linked) and PFA(c-linked) are each equiconsistent with the existence of a $\Sigma _{1}^{2}$ -indescribable cardinal. Our upper bound for SPFA(c-c.c.) is a $\Sigma _{2}^{2}$ -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c⁺-linked) is a cardinal κ that is $(\kappa ^{+},\Sigma _{1}^{2})$ -subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(c) is a consequence of SPFA(c⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(c⁺⁺-c.c.) is a cardinal κ that is $(\kappa ^{+},\Sigma _{2}^{2})$ -subcompact, which is also strictly weaker than κ⁺-supercompact

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Citations of this work

Resurrection axioms and uplifting cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.
Global square sequences in extender models.Martin Zeman - 2010 - Annals of Pure and Applied Logic 161 (7):956-985.

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References found in this work

The lottery preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
Square in core models.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.
Semiproper forcing axiom implies Martin maximum but not PFA+.Saharon Shelah - 1987 - Journal of Symbolic Logic 52 (2):360-367.

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