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PreprintCC BY 4.0arXiv

Vopěnka's Principle without Choice: Preservation under Symmetric Extensions

  • Tom de Groot
  • Wojciech Aleksander Wołoszyn
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Abstract

We give a detailed proof that every set-sized symmetric extension of a model of ZF+VP again satisfies ZF+VP, where VP is formulated for arbitrary set-sized languages. For each standard n≥ 1, the same preservation theorem holds for VP(Πn), with set parameters and arbitrary set-sized languages. We also prove that ZF+VP and ZFC+VP are equiconsistent, and that ZF+VP proves the existence of a proper class of Löwenheim-Skolem cardinals. Relative to Con(ZF+VP), the axiom DC is independent of ZF+VP+¬AC. The preservation theorem also yields Feferman-Lévy and full Solovay models satisfying VP. We consider whether every countable model of ZF+VP has a class-generic extension satisfying ZFC+VP.

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Cite

BibTeXRIS
BibTeX
@misc{groot2026vopěnkas,
  title = {{Vopěnka's Principle without Choice: Preservation under Symmetric Extensions}},
  author = {Tom de Groot and Wojciech Aleksander Wołoszyn},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.06856},
  eprint = {2609.06856},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06856},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Vopěnka's Principle without Choice: Preservation under Symmetric Extensions
AU  - Tom de Groot
AU  - Wojciech Aleksander Wołoszyn
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.06856
UR  - https://arxiv.org/abs/2609.06856
AB  - We give a detailed proof that every set-sized symmetric extension of a model of ZF+VP again satisfies ZF+VP, where VP is formulated for arbitrary set-sized languages. For each standard n≥ 1, the same preservation theorem holds for VP(Π_n), with set parameters and arbitrary set-sized languages. We also prove that ZF+VP and ZFC+VP are equiconsistent, and that ZF+VP proves the existence of a proper class of Löwenheim-Skolem cardinals. Relative to Con(ZF+VP), the axiom DC is independent of ZF+VP+¬AC. The preservation theorem also yields Feferman-Lévy and full Solovay models satisfying VP. We consider whether every countable model of ZF+VP has a class-generic extension satisfying ZFC+VP.
ER  -