Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum

Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Lowenheim–Skolem Theorems (SDLSs) of stationary logic and their variations. In Fuchino (2020) it has been shown that SDLS for ordinary with weak second-order parameters $$\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)$$ down to $${<}\,\aleph _2$$ is equivalent conjunction CH Cox’s Diagonal Reflection Principle internally clubness. We show without {SDLS}^-({\mathcal _0}_{stat},{<}\,2^{\aleph _0})$$ $${<}\,2^{\aleph _0}$$ implies size continuum $$\aleph . contrast, an internal interpretation can satisfy under being $$>\aleph This be version set also consider a this {SDLS}^{int}_+({\mathcal {L}}^{PKL}_{stat},{<}\,2^{\aleph reflection consistent assumption consistency ZFC $$+$$ “the existence supercompact cardinal” (at least) weakly Mahlo. These three “axioms” terms are consequences instances strengthening generic supercompactness which call Laver-generic supercompactness. Existence cardinal each these fixes cardinality _1$$ or very large respectively. one cardinals “ $$++$$ ” corresponding forcing axiom.