Archive for Mathematical Logic 60 (3):495-523 (2021)

Abstract
Continuing, we study the Strong Downward Löwenheim–Skolem Theorems of the stationary logic and their variations. In Fuchino et al. it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters \. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size \. We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation \\) for reflection down to \ is consistent under the assumption of the consistency of ZFC \ “the existence of a supercompact cardinal” and this SDLS implies that the continuum is 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 \ or \ 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.
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DOI 10.1007/s00153-020-00751-6
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References found in this work BETA

On the Existence of Skinny Stationary Subsets.Yo Matsubara, Hiroshi Sakai & Toshimichi Usuba - 2019 - Annals of Pure and Applied Logic 170 (5):539-557.
Laver Sequences for Extendible and Super-Almost-Huge Cardinals.Paul Corazza - 1999 - Journal of Symbolic Logic 64 (3):963-983.
Laver Sequences for Extendible and Super-Almost-Huge Cardinals.Paul Corazza - 1999 - Journal of Symbolic Logic 64 (3):963-983.

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