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PreprintCC BY 4.0arXiv

Set-theoretic inevitability

  • Chris Scambler
  • Cesare Straffelini
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Abstract

After defining the concept of inevitability for statements with the modal logic of forcing, we raise the question of which statements are inevitable under ZFC and under large cardinal hypotheses. We show that SOA (defined as ZFC- + V=HC) plus the Σ12-perfect set property is ZFC-inevitable, and that Projective Determinacy is inevitable in the theory of ZFC with a proper class of Woodin cardinals. We then discuss the possible conceptual significance of these results for the foundations of set theory.

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Cite

BibTeXRIS
BibTeX
@misc{scambler2026settheoretic,
  title = {{Set-theoretic inevitability}},
  author = {Chris Scambler and Cesare Straffelini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31040},
  eprint = {2610.31040},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31040},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Set-theoretic inevitability
AU  - Chris Scambler
AU  - Cesare Straffelini
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31040
UR  - https://arxiv.org/abs/2610.31040
AB  - After defining the concept of inevitability for statements with the modal logic of forcing, we raise the question of which statements are inevitable under ZFC and under large cardinal hypotheses. We show that SOA (defined as ZFC^- + V=HC) plus the Σ^1_2-perfect set property is ZFC-inevitable, and that Projective Determinacy is inevitable in the theory of ZFC with a proper class of Woodin cardinals. We then discuss the possible conceptual significance of these results for the foundations of set theory.
ER  -