{
  "id": "1512.05122",
  "version": "v1",
  "published": "2015-12-16T10:47:08.000Z",
  "updated": "2015-12-16T10:47:08.000Z",
  "title": "A Uniform Characterization of $Σ_1$-Reflection over the Fragments of Peano Arithmetic",
  "authors": [
    "Anton Freund"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that the theory $I\\Sigma_1$ of $\\Sigma_1$-induction proves the following statement: For all $n\\geq 2$, the uniform $\\Sigma_1$-reflection principle over the theory $I\\Sigma_n$ is equivalent to the totality of the function $F_{\\omega_n}$ at stage $\\omega_n$ of the fast-growing hierarchy. The method applied is a formalization of infinite proof theory. The literature contains several proofs which place the quantification over $n$ in the meta-theory (and also prove the separate cases $n=0,1$). In contrast, the author knows of no explicit argument that would allow us to internalize the quantification while keeping the meta-theory as low as $I\\Sigma_1$. It is well possible that this has been considered before. Our aim is merely to provide a detailed exposition of this important result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-16T10:47:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F30"
    ],
    "keywords": [
      "peano arithmetic",
      "uniform characterization",
      "infinite proof theory",
      "explicit argument",
      "reflection principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151205122F"
    }
  }
}