Caicedo–Veličković conjecture under Martin’s Maximum

Establish that if W ⊆ V are transitive models of ZFC + Martin’s Maximum (MM) with Card^W = Card^V, then W and V have the same ω1-sequences of ordinals.

Background

The conjecture arises from results showing strong rigidity of models under forcing axioms (e.g., BPFA implies that sharing ω2 ensures P(ω1) ⊆ W). It proposes that under MM, sharing the class of cardinals forces agreement on all ω1-sequences of ordinals.

Within the paper, the conjecture is used to motivate investigation into cardinal-preserving embeddings: under MM, the conjecture would imply that any cardinal-preserving embedding j: M → N with M ⊆ V must be the identity, providing a powerful obstruction to such embeddings.