Hierarchies of forcing axioms I

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


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



    Upload a copy of this work     Papers currently archived: 91,088

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Hierarchies of Forcing Axioms II.Itay Neeman - 2008 - Journal of Symbolic Logic 73 (2):522 - 542.
Projective Well-Orderings and Bounded Forcing Axioms.Andrés Eduardo Caicedo - 2005 - Journal of Symbolic Logic 70 (2):557 - 572.
A maximal bounded forcing axiom.David Asperó - 2002 - Journal of Symbolic Logic 67 (1):130-142.
Simple forcing notions and forcing axioms.Andrzej Rosłanowski & Saharon Shelah - 1997 - Journal of Symbolic Logic 62 (4):1297-1314.
Proper forcing and l(ℝ).Itay Neeman & Jindřich Zapletal - 2001 - Journal of Symbolic Logic 66 (2):801-810.
PFA and Ideals on $\omega_{2}$ Whose Associated Forcings Are Proper.Sean Cox - 2012 - Notre Dame Journal of Formal Logic 53 (3):397-412.
Forcing Indestructibility of Set-Theoretic Axioms.Bernhard König - 2007 - Journal of Symbolic Logic 72 (1):349 - 360.
Small forcing makes any cardinal superdestructible.Joel David Hamkins - 1998 - Journal of Symbolic Logic 63 (1):51-58.


Added to PP

35 (#408,502)

6 months
5 (#311,051)

Historical graph of downloads
How can I increase my downloads?

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.

Add more citations

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.

Add more references