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

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 $\mathrm{PFA}_{\aleph _3}$.