The ABCD hypothesis — Draconitic Research

Abstract

If α and β are cardinals, then the cardinal exponentiation αβ is the set of all functions from α into β. The ABCD hypothesis is the statement that for all infinite cardinals α≤β and γ≤δ, |αβ|≤|γδ| if and only if α≤γ and β≤δ. The Axiom of Determinacy, AD, states that every two-player infinite game on ω has a winning strategy for one of the two players. Let Θ be the supremum of the ordinals onto which R surjects. This paper will show that if AD holds and V=L(R), then the ABCD hypothesis holds below Θ.

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BibTeX RIS

BibTeX

@article{chan2026abcd,
  title = {{The ABCD hypothesis}},
  author = {William Chan},
  journal = {Proceedings of the National Academy of Sciences of the United States of America},
  year = {2026},
  month = oct,
  doi = {10.1073/pnas.2621223123},
  url = {https://doi.org/10.1073/pnas.2621223123},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - JOUR
TI  - The ABCD hypothesis
AU  - William Chan
PY  - 2026
DA  - 2026/10/29
JO  - Proceedings of the National Academy of Sciences of the United States of America
DO  - 10.1073/pnas.2621223123
UR  - https://doi.org/10.1073/pnas.2621223123
AB  - If α and β are cardinals, then the cardinal exponentiation αβ is the set of all functions from α into β. The ABCD hypothesis is the statement that for all infinite cardinals α≤β and γ≤δ, |αβ|≤|γδ| if and only if α≤γ and β≤δ. The Axiom of Determinacy, AD, states that every two-player infinite game on ω has a winning strategy for one of the two players. Let Θ be the supremum of the ordinals onto which R surjects. This paper will show that if AD holds and V=L(R), then the ABCD hypothesis holds below Θ.
SN  - 0027-8424
SN  - 1091-6490
ER  -

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