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

Abstract

We introduce and study the first-order Generic Vopĕnka's Principle, which states that for every definable proper class of structures $$\mathcal {C}$$C of the same type, there exist $$B\ne A$$BźA in $$\mathcal {C}$$C such that B elementarily embeds into A in some set-forcing extension. We show that, for $$n\ge 1$$nź1, the Generic Vopĕnka's Principle fragment for $$\Pi _n$$źn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for $$n\in \omega $$nźź, which we introduce here, is a natural generic analogue for the $$C^{(n)}$$C(n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopĕnka'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, $$\mathrm{wPFA}$$wPFA. The axiom $$\mathrm{wPFA}$$wPFA states that for every transitive model $$\mathcal M$$M in the language of set theory with some $$\omega _1$$ź1-many additional relations, if it is forced by a proper forcing $$\mathbb P$$P that $$\mathcal M$$M satisfies some $$\Sigma _1$$Σ1-property, then V has a transitive model $$\bar{\mathcal M}$$M�, satisfying the same $$\Sigma _1$$Σ1-property, and in some set-forcing extension there is an elementary embedding from $$\bar{\mathcal M}$$M� into $$\mathcal M$$M. This is a weakening of a formulation of $$\mathrm{PFA}$$PFA due to Claverie and Schindler (J Symb Logic 77(2):475---498, 2012), which asserts that the embedding from $$\bar{\mathcal M}$$M� to $$\mathcal M$$M exists in V. We show that $$\mathrm{wPFA}$$wPFA is equiconsistent with a remarkable cardinal. Furthermore, the axiom $$\mathrm{wPFA}$$wPFA implies $$\mathrm{PFA}_{\aleph _2}$$PFAź2, the Proper Forcing Axiom for antichains of size at most $$\omega _2$$ź2, but it is consistent with $$\square _\kappa $$źź for all $$\kappa \ge \omega _2$$źźź2, and therefore does not imply $$\mathrm{PFA}_{\aleph _3}$$PFAź3.