ABCD Conjecture Under Determinacy

Definitions and mathematical statement

Work in $\mathrm{ZF}+\mathrm{AD}$, where $\mathrm{ZF}$ is Zermelo--Fraenkel set theory without the axiom of choice, and $\mathrm{AD}$ asserts that every perfect-information game of length $\omega$, with moves in ${\mathbb{N}}$ and a prescribed payoff set $A\subseteq{\mathbb{N}}^{\mathbb{N}}$, is determined. Define \[ \Theta=\sup\{\xi\in\mathrm{Ord}:\text{there exists a surjection }{\mathbb{R}}\twoheadrightarrow\xi\}. \] For sets $X,Y$, write $|X|\leq|Y|$ when there is an injection $X\hookrightarrow Y$. Interpret the exponentiation in the supplied statement as function-set exponentiation: \[ \alpha^\beta=\{f\mid f:\beta\longrightarrow\alpha\}. \] This interpretation requires confirmation from the cited report, since the supplied excerpt does not define the notation.

The supplied formulation of the ABCD conjecture asks whether, for every quadruple of ordinals $\alpha,\beta,\gamma,\delta<\Theta$ satisfying $\beta\leq\alpha$ and $\delta\leq\gamma$, one has \[ |\alpha^\beta|\leq|\gamma^\delta| \quad\Longleftrightarrow\quad \left(|\beta|\leq|\delta|\ \text{and}\ |\alpha|\leq|\gamma|\right). \] Unresolved qualification. The supplied quantifiers include zero and all finite ordinals, and no infinitude hypotheses are stated. Under the function-set interpretation, the displayed equivalence fails already for $(\alpha,\beta,\gamma,\delta)=(3,1,2,2)$: the left side compares sets of sizes $3$ and $4$, whereas the right side requires $3\leq2$. Thus the supplied unrestricted formulation cannot serve as a valid conjecture without clarification of its domain or another missing qualification. No replacement hypotheses are imposed here.

Short English statement

The supplied ABCD conjecture asks whether, under $\mathrm{ZF}+\mathrm{AD}$, comparison of two function-set cardinalities below $\Theta$ is equivalent to comparing their bases and exponents separately. Its stated domain admits finite counterexamples, so the intended hypotheses require clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006171$, title "ABCD Conjecture Under Determinacy," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year 2025 but no named proposer.
• Supplied primary-source citation: Set Theory (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/2, pp. 75--140. https://doi.org/10.4171/owr/2025/2.
• Supplied literature source: "Oberwolfach Report 2/2025." https://content.ems.press/assets/public/full-texts/serials/owr/22/1/14299082/online/10.4171-owr-2025-2.pdf.
• Literature assessment dated 2026-08-22: the supplied curation labels the conjecture open under $\mathrm{ZF}+\mathrm{AD}$ and reports a proof under the stronger $\mathrm{AD}^{+}$ hypothesis. The retained contextual excerpt also reports special cases under $\mathrm{ZF}+\mathrm{AD}$, without specifying them. These are supplied dated curation and source claims, not independently verified assessments; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. This is supplied verification metadata, not verification performed for this draft, and it does not resolve the formulation issue identified above.