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Does the Partition Principle imply the Axiom of Choice?

Theorem 1.1: Con(ZF) implies Con(ZF + PP + ACwo + $\neg$ AC). Theorem 1.2: over every countable transitive model $V$ of ZFC there are a set forcing and a transitive symmetric submodel $W$ with $V\subseteq W\subseteq V[G]$ , the same ordinals, $W\models$ ZF + PP + ACwo + $\neg$ AC, and no new countable sequences of elements of $V$ . The construction makes every non-wellorderable image of a fixed set bijective with it and uses Ryan-Smith's local-to-global reduction. Remarks draw consequences for the dual Cantor-Schroeder-Bernstein principle and the Weak Partition Principle. It does not settle whether the Weak Partition Principle implies AC or PP, nor other cardinal-comparison principles beyond those remarks.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Does the Partition Principle imply the Axiom of Choice? · Partition Principle vs Choice

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Does the Partition Principle imply the Axiom of Choice?

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No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the manuscript were read against the question; Theorem 1.1 (if ZF is consistent, so is ZF + PP + ACwo + not AC) answers it as posed. lean/formalization.yaml lists PartitionPrinciple (declaration ...TransitiveGround.exists_model_partitionPrinciple_without_choice, file OAI/SetTheory/PartitionPrinciple/Models/Separation.lean). Its comparator statement was read: it assumes a countable transitive ground with an internal strongly inaccessible cardinal and yields a transitive ZF model of PP with Choice failing, which is narrower than the paper's Theorem 1.2 (no inaccessible, and also ACwo and no new countable sequences). A second challenge, ComparatorChallenges/PartitionConsistency.json, is not in the formalization catalogue; its solution module OAI/SetTheory/PartitionConsistency/Main.lean exists at the pinned commit, and its statement was read here: syntactic Con(ZF) implies Con(ZF + PP + ACwo + not AC) for an explicit first-order natural deduction over the pure membership language. That is the headline. Neither was rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.