{
  "id": "2010.01922",
  "version": "v1",
  "published": "2020-10-05T11:19:03.000Z",
  "updated": "2020-10-05T11:19:03.000Z",
  "title": "Forcing axioms and the complexity of non-stationary ideals",
  "authors": [
    "Sean Cox",
    "Philipp Lücke"
  ],
  "comment": "33 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\\omega_2$ to sets of ordinals of countable cofinality is $\\Delta_1$-definable by formulas with parameters in $H(\\omega_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S. Friedman, Wu and Zdomskyyshow by showing that the $\\Delta_1$-definability of the non-stationary ideal on $\\omega_2$ is compatible with arbitrary large values of the continuum function at $\\omega_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-05T11:19:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E57",
      "03E35",
      "03E47"
    ],
    "keywords": [
      "complexity",
      "strong forcing axioms",
      "full non-stationary ideal",
      "arbitrary large values",
      "martins maximum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}