{
  "id": "2203.04186",
  "version": "v1",
  "published": "2022-03-08T16:25:28.000Z",
  "updated": "2022-03-08T16:25:28.000Z",
  "title": "The Special Tree Number",
  "authors": [
    "Corey Bacal Switzer"
  ],
  "comment": "17 pages, 1 figure, submitted",
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce a new cardinal characteristic called the {\\em special tree number}, denoted $\\mathfrak{st}$, defined to be the least size of a tree of height $\\omega_1$ which is neither special nor has a cofinal branch. Classical facts imply that $\\aleph_1 \\leq \\mathfrak{st} \\leq 2^{\\aleph_0}$, $\\mathsf{MA}$ implies $\\mathfrak{st} = 2^{\\aleph_0}$ while $\\mathfrak{st} = \\aleph_1$ is consistent with $\\mathsf{MA}({\\rm Knaster}) + 2^{\\aleph_0} = \\kappa$ for any regular $\\kappa$ thus the value of $\\mathfrak{st}$ is not decided by $\\mathsf{ZFC}$ and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that $\\mathfrak{st} = 2^{\\aleph_0} = \\kappa$ for any regular $\\kappa$ while ${\\rm non}(\\mathcal M) = \\mathfrak{a} = \\mathfrak{s} = \\mathfrak{g} = \\aleph_1$. In particular $\\mathfrak{st}$ is independent of the lefthand side of Cicho\\'{n}'s diagram, amongst other things. The proof involves an in depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T16:25:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "special tree number",
      "cardinal characteristic",
      "standard ccc forcing notion",
      "independent interest",
      "aronszajn trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}