{
  "id": "1605.06051",
  "version": "v1",
  "published": "2016-05-19T16:59:59.000Z",
  "updated": "2016-05-19T16:59:59.000Z",
  "title": "$\\mathbf{Σ_{3}^{1}}$ Generic Absoluteness, Measurability and Perfect Set Theorems for Equivalence Relations with small classes",
  "authors": [
    "Ohad Drucker"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce a regularity property of equivalence relations - the existence of a perfect set of pairwise inequivalent elements when all equivalence classes are $I$ - small with respect to a $\\sigma$ ideal $I$. We show this regularity property for $\\mathbf{\\Sigma^1_2}$ and $\\mathbf{\\Delta^1_2}$ equivalence relations is interconnected with $\\Sigma^1_3$ generic absoluteness on one hand and transcendence over $L$ on the other hand. For example, we show that given a definable enough provably ccc $\\sigma$ - ideal $I$, all $\\mathbf{\\Delta^1_2}$ equivalence relations with $I$ - small classes have a perfect set of pairwise inequivalent elements if and only if there is $\\Sigma^1_3$ $\\mathbb{P}_I$ generic absoluteness, if and only if for every real $z$ and $B$ positive there is a $\\mathbb{P}_I$ generic real over $L[z]$ in $B$. For the case of the meager ideal, we prove that $\\mathbf{\\Sigma^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if for every real $z$ there is a Cohen real over $L[z]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-19T16:59:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivalence relations",
      "generic absoluteness",
      "perfect set theorems",
      "small classes",
      "pairwise inequivalent elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}