{
  "id": "2009.03348",
  "version": "v1",
  "published": "2020-09-07T18:08:36.000Z",
  "updated": "2020-09-07T18:08:36.000Z",
  "title": "Strong downward Löwenheim-Skolem theorems for stationary logics, II -- reflection down to the continuum",
  "authors": [
    "Sakaé Fuchino",
    "André Ottenbreit Maschio Rodrigues",
    "Hiroshi Sakai"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Continuing the previous paper, we study the Strong Downward L\\\"owenheim-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 $<\\aleph_2$ 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 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 ${\\cal P}_\\kappa\\lambda$ version of the stationary logic and show that the SDLS for this logic in internal interpretation 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-07T18:08:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E50",
      "03E55",
      "03E65"
    ],
    "keywords": [
      "stationary logic",
      "strong downward löwenheim-skolem theorems",
      "weak second-order parameters",
      "generic large cardinals implies",
      "supercompact cardinal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}