Hierarchies of Forcing Axioms II

Journal of Symbolic Logic 73 (2):522 - 542 (2008)
  Copy   BIBTEX


A $\Sigma _{1}^{2}$ truth for λ is a pair 〈Q, ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ Hλ+ such that (Hλ+; ε, B) $(H_{\lambda +};\in ,B)\vDash \psi [Q]$ . A cardinal λ is $\Sigma _{1}^{2}$ indescribable just in case that for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $\overline{\lambda}<\lambda $ so that $\overline{\lambda}$ is a cardinal and $\langle Q\cap H_{\overline{\lambda}},\psi \rangle $ is a $\Sigma _{1}^{2}$ truth for $\overline{\lambda}$ . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is $\Sigma _{1}^{2}$ indescribable if for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $??\leq \overline{\lambda}<\kappa,??\subseteq H_{\overline{\lambda}}$ , and π: $H_{\overline{\lambda}}\rightarrow H_{\lambda}$ so that $??$ is a cardinal, $\langle ??,\psi \rangle $ is a $\Sigma _{1}^{2}$ truth for $??$ , and π is elementary from $(H_{\overline{\lambda}};\in,??,??)$ with $\pi \,|\,??={\rm id}$ . We prove that the restriction of the proper forcing axiom to c-linked posets requires a $\Sigma _{1}^{2}$ indescribable cardinal in L, and that the restriction of the proper forcing axiom to c⁺-linked posets, in a proper forcing extension of a fine structural model, requires a $\Sigma _{1}^{2}$ indescribable 1-gap [κ, κ⁺]. These results show that the respective forward directions obtained in " Hierarchies of Forcing Axioms I" by Neeman and Schimmerling are optimal



    Upload a copy of this work     Papers currently archived: 91,164

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 I.Itay Neeman & Ernest Schimmerling - 2008 - Journal of Symbolic Logic 73 (1):343-362.
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

25 (#592,433)

6 months
7 (#328,545)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

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

Combinatorial principles in the core model for one Woodin cardinal.Ernest Schimmerling - 1995 - Annals of Pure and Applied Logic 74 (2):153-201.
Square in core models.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.
The covering lemma for K.Tony Dodd & Ronald Jensen - 1982 - Annals of Mathematical Logic 22 (1):1-30.
Characterization of □κin core models.Ernest Schimmerling & Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):1-72.

Add more references