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Math-589-A2 · revision r1 · on Math-589

OpenAI claim summary for Math-589

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 244; 2026-09-24).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Partition Principle Implies Axiom of Choice
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Partition Principle Implies Axiom of Choice

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 244; 2026-09-24), the asserted result is that relative to consistency of ZF, the Partition Principle can hold together with ordinal-indexed choice and failure of full Choice.

Mathematical statement of the claimed result

Manuscript 1.
The Partition Principle $\mathrm{PP}$ means that a surjection $X\twoheadrightarrow Y$ implies some injection $Y\hookrightarrow X$, not necessarily a section. Let $\mathrm{AC}_{\rm wo}$ be choice for every ordinal-indexed family of nonempty sets. The claimed relative consistency is \[ \operatorname{Con}(\mathrm{ZF})\Rightarrow \operatorname{Con}(\mathrm{ZF}+\mathrm{PP}+\mathrm{AC}_{\rm wo}+\neg\mathrm{AC}). \] More concretely, from a countable transitive $V\models\mathrm{ZFC}$ the manuscript claims set forcing and a transitive symmetric intermediate model $V\subseteq W\subseteq V[G]$ with the same ordinals and $W\models\mathrm{ZF}+\mathrm{PP}+\mathrm{AC}_{\rm wo}+\neg\mathrm{AC}$. Every countable sequence of elements of $V$ that lies in $W$ already lies in $V$. Thus PP is claimed not to imply full Choice over ZF at this relative-consistency strength.

Outline of the argument

For manuscript 1 (family 244; 2026-09-24), a symmetric forcing construction creates nonwellorderable generic labels and orbit maps that present arbitrary sets by label--ordinal products. Controlled supports give ordinal-family choices, while quotient classification supplies injections opposite surjections without making the labels wellorderable.

Formal verification

For manuscript 1 (family 244; 2026-09-24), the documented Lean scope is the relative-consistency implication for ZF, the Partition Principle, ordinal-indexed choice, and failure of Choice; a separate transitive-model construction has an inaccessible-cardinal ground assumption and does not encode all the paper's preservation assertions.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are PartitionConsistency.lean, PartitionPrinciple.lean.

Their named declarations include OAI.PartitionConsistency.main_consistency, OAI.PartitionPilot.Forcing.TransitiveGround.exists_model_partitionPrinciple_without_choice.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources