Skip to content
DraconiticResearch
Search Sources API
190,869 studies31 sourcesUpdated 25 min ago
PreprintarXiv licencearXiv

RCA0 Proves the Collapse of the Intersection Principles Hierarchy

  • Patrizio Cintioli
Open at sourcePDFDOICite

Abstract

This paper shows that the pairwise intersection principle (2IP) implies, over RCA0, the existence of a Cohen $1$-generic real relative to every given set. Together with the known implications, this yields the equivalence over RCA0 of the finite intersection principle (FIP), 2IP, and $1$-GEN. In fact, FIP is equivalent to nIP for every fixed n≥ 2. These results give affirmative answers to Questions~1 and~2 of Cholak, Downey, and Igusa, and to Question~9.10.26 of Dzhafarov and Mummert. The proof uses cumulative, nested supports satisfying a finite Helly property and a decoding argument based on a least selected label. Neither Σ02 induction nor Σ02 bounding is required.

Related

  1. Extension of valuation rings as limits of complete intersection algebras
    Dorin Popescu
    arXivPreprintCC0Full textAlgebra & number theoryarXiv
  2. Counting curves with bounded intersection
    Hugo Parlier
    arXivPreprintarXiv licenceFull textDynamical systemsGeometry & topologyarXiv
  3. Intersection Orbital Graphs of Permutation Groups and Reconstruction of Simple Groups
    Shahram Mehry
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Good stratifications of non-degenerate locally tame complete intersection varieties
    Kazumasa Inaba
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. The Intersection of nested filled-in little Julia sets have zero area
    Carsten Lunde Petersen
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{cintioli2026rca0,
  title = {{$\mathsfRCA_0$ Proves the Collapse of the Intersection Principles Hierarchy}},
  author = {Patrizio Cintioli},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01571},
  eprint = {2611.01571},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01571},
}
RIS
TY  - UNPB
TI  - RCA_0 Proves the Collapse of the Intersection Principles Hierarchy
AU  - Patrizio Cintioli
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01571
UR  - https://arxiv.org/abs/2611.01571
AB  - This paper shows that the pairwise intersection principle (2IP) implies, over RCA_0, the existence of a Cohen $1$-generic real relative to every given set. Together with the known implications, this yields the equivalence over RCA_0 of the finite intersection principle (FIP), 2IP, and $1$-GEN. In fact, FIP is equivalent to nIP for every fixed n≥ 2. These results give affirmative answers to Questions~1 and~2 of Cholak, Downey, and Igusa, and to Question~9.10.26 of Dzhafarov and Mummert. The proof uses cumulative, nested supports satisfying a finite Helly property and a decoding argument based on a least selected label. Neither Σ^0_2 induction nor Σ^0_2 bounding is required.
ER  -