{
  "id": "rigidity-of-the-turing-degrees",
  "title": "The rigidity of the Turing degrees",
  "description": "Computability relative to a set, defined by computably enumerable sets of triples; joins and Turing degrees; real numbers as cuts. Meager sets, Baire's theorem, continuity of Borel functions on comeager sets, and the avoidance lemma. Recovery of an irrational number from four values of a Borel function with countable fibres that is compatible with sums. The rigidity theorem for the Turing degrees, from the theorem of Slaman and Woodin on Borel representations of automorphisms, which is a named hypothesis. The course is an exposition of OpenAI's proof (2026) of the rigidity of the Turing degrees.",
  "units": [
    {
      "id": "relative-computability-and-the-turing-degrees",
      "title": "Relative computability and the Turing degrees",
      "reader": "relative-computability-and-the-turing-degrees.html",
      "source": "src/relative-computability-and-the-turing-degrees.md",
      "source_sha256": "9bcd27fbf90c21022cb2b054ff3bdf10a3c44b88b4266dd486e07446e61b27a4",
      "status": "author_self_checked"
    },
    {
      "id": "category-and-computability",
      "title": "Category and computability",
      "reader": "category-and-computability.html",
      "source": "src/category-and-computability.md",
      "source_sha256": "cd0a3b97c6ee4a5b3193ec8dc286ae0652ce78d85dcaa7e639a74edba5aea8e9",
      "status": "author_self_checked"
    },
    {
      "id": "recovery-from-four-values",
      "title": "Recovery from four values",
      "reader": "recovery-from-four-values.html",
      "source": "src/recovery-from-four-values.md",
      "source_sha256": "5499ed641a7c3b8ff5621cdbe0e6028642052c9e9f560ba7efa266b91ebc2201",
      "status": "author_self_checked"
    },
    {
      "id": "the-rigidity-theorem",
      "title": "The rigidity theorem",
      "reader": "the-rigidity-theorem.html",
      "source": "src/the-rigidity-theorem.md",
      "source_sha256": "c3c26795dea364bb3e1c625a113b87d0e9fb842e42dd93cc865a71d4fcfc1a64",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Mathematical Logic, Set Theory, and Computability (C80): computable functions, computable and computably enumerable sets, the list W_e of the computably enumerable sets (Open Logic Project, chapter Computability Theory)",
    "core course Measure and Integration (D10): Borel sets; the topology of the line and the plane",
    "the lessons prove every other fact they use, except the named hypothesis (SW) of the fourth lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}