Archive for Mathematical Logic

, Volume 56, Issue 1–2, pp 1–20 | Cite as

Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

Article

Abstract

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).

Keywords

Large cardinals Vopěnka’s Principle Generic Vopěnka’s Principle Remarkable cardinals Proper Forcing Axiom 

Mathematics Subject Classification

03E35 03E55 03E57 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Joan Bagaria
    • 1
    • 2
  • Victoria Gitman
    • 3
  • Ralf Schindler
    • 4
  1. 1.ICREABarcelonaSpain
  2. 2.Departament de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain
  3. 3.Mathematics Program, CUNY Graduate CenterThe City University of New YorkNew YorkUSA
  4. 4.Institut für mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und InformatikUniversität MünsterMünsterGermany

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