{
  "abstract": "For various reasons, higher-order objects are often studied in mathematical logic via second-order `codes' or `representations'. An important example hailing from real analysis and topology is provided by open sets, which are represented by unions $\\cup_{n\\in \\mathbb{N}}I_{n}$ where each $I_{n}$ is a basic open interval. Now, Montalbán has recently highlighted the importance of robustness of logical systems in Reverse Mathematics (abbreviated RM). It is then a natural RM-question whether basic properties of open sets are robust under slight modifications of the coding of open sets. Here, we study the representation where $I_{n}$ as above is \\emph{either} an open interval \\emph{or} the union of two such intervals \\emph{but} we cannot decide which one. Under this slight variation of the usual coding, basic properties of open and closed sets readily imply the relatively strong system ATR$_{0}$ from RM. Moreover, we obtain equivalences for the former properties and the \\emph{enumeration principle}. The latter states that countable sets can be enumerated and boasts many equivalences from Fourier analysis. Along the way, we investigate the RM-properties of the closure, interior, and boundary of sets of reals, a study interesting in its own right.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18371",
  "author_total": 1,
  "authors": [
    {
      "name": "Sam Sanders"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2z7nf4bjd6y83.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18371",
  "fulltext_url": null,
  "id": "2z7nf4bjd6y83",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18371",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7213527444",
  "pdf_url": "https://arxiv.org/pdf/2609.18371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2z7nf4bjd6y83.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Coding is non-robust",
  "title_text": "Coding is non-robust",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/2z7nf4bjd6y83",
  "venue": "arXiv",
  "versions": []
}