{
  "abstract": "Let F(x,n) denote the formula ∃ab ∀i⩽n ∃swpq ∀jv ∃eg {(s+w)^2+3w+s=2i ∧ ⟨[j=w ∨ v=q] ∨ [j=3i ∧ v=p+q] ∨ [j=s ∧ (v=p ∨ (i=n ∧ v=q+x))] ∨ [j=3i+1 ∧ v=pq] ⇒ a=v+e+ejb ∧ v+g=b⟩} from J. P. Jones' article in vol. 43 of J. Symbolic Logic. From the results of Jones' article, it follows that the set J={n∈N: ¬F(n,n)} is co-recursively enumerable and not recursively enumerable. We prove that the set W={n∈N: ∃p,q∈N ((2n=(p+q)(p+q+1)+2q) ∧ ∀(x_0,...,x_p)∈N^{p+1} ∃(y_0,...,y_p)∈{0,...,q}^{p+1} ((∀i,j,k∈{0,...,p} (x_j+1=x_k ⇒ y_j+1=y_k)) ∧ (∀i,j,k∈{0,...,p} (x_i \\cdot x_j=x_k ⇒ y_i \\cdot y_j=y_k))))} is co-recursively enumerable and not recursively enumerable. Let β:N^3→N denote Gödel's β function. For x_1,x_2,x_3∈N, β(x_1,x_2,x_3) equals the remainder after integer division of x_1 by 1+(x_3+1) \\cdot x_2. We prove that the set W consists of all n∈N such that ∀u,v∈N ∃a,b,p,q∈N ((2n=(p+q)(p+q+1)+2q) ∧ ∀i,j,k∈{0,...,p} ((β(a,b,i)⩽q) ∧ (β(u,v,j)+1=β(u,v,k) ⇒ β(a,b,j)+1=β(a,b,k)) ∧ (β(u,v,i) \\cdot β(u,v,j)=β(u,v,k) ⇒ β(a,b,i) \\cdot β(a,b,j)=β(a,b,k)))). We express the above formula in Peano arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Apoloniusz Tyszka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atdqs7qnwzmwz.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "atdqs7qnwzmwz",
  "issns": [],
  "landing_url": "https://hal.science/hal-01614087v85",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-lo"
    ],
    [
      "hal",
      "1.math.math-nt"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-01614087/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atdqs7qnwzmwz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:35Z",
      "native_id": "hal-01614087",
      "source": "hal",
      "url": "https://hal.science/hal-01614087v85"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-nt",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-lo",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A non-recursively enumerable set W⊆N, where N\\W is recursively enumerable and the predicate n∈W expresses a fundamental property of n",
  "title_text": "A non-recursively enumerable set W⊆N, where N\\W is recursively enumerable and the predicate n∈W expresses a fundamental property of n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:35Z",
  "url": "https://research.draconitic.com/paper/atdqs7qnwzmwz",
  "venue": "HAL",
  "versions": []
}