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

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures C of the same type, there exist B 6= A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka’s Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for n ∈ ω, 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 [1]. 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, wPFA, which states that for every transitive modelM in the language of set theory with some ω1-many additional relations, if it is forced by a proper forcing P thatM satisfies some Σ1-property, then V has a transitive model M̄, which satisfies the same Σ1property, and in some set-forcing extension there is an elementary embedding from M̄ intoM. This is a weakening of a formulation of PFA due to Schindler and Claverie [2], which asserts that the embedding from M̄ toM exists in V . We show that wPFA is equiconsistent with a remarkable cardinal and that wPFA implies PFAא2 , the proper forcing axiom for antichains of size at most ω2, but it is consistent with κ for all κ ≥ ω2, and therefore does not imply PFAא3 .