2020/09/07 by Fuchino, Sakaé, Rodrigues, André Ottenbreit Maschio, Sakai, Hiroshi · 1 citation
#03E35 #03E50 #03E55 #03E65 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2009.03348
Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to ℵ2. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size <2ℵ0. We also consider a \cal Pκλ version of the stationary logic and show that the SDLS for this logic in internal interpretation for reflection down to <2ℵ0 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 ℵ1 or ℵ2 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.