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

Abstract

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