Result 244, Mathematical logic

The Partition Principle does not imply Choice

Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Comparing the sizes of sets can look like a way to choose elements from them. The manuscript reports a precise separation between these ideas, clarifying which assumptions are needed to make arbitrary choices in set theory.

What changes?

The Partition Principle says that whenever a map from one set onto another hits every target, the target admits a one-to-one map back. Assuming ZF, set theory without Choice, is consistent, the manuscript reports consistency with this principle, Choice for ordinal-indexed families, and failure of full Choice. Separately, from any countable transitive model of ZFC (ZF with Choice), it constructs a transitive symmetric extension with these properties, the same ordinals, and no new countable sequences of ground-model elements.

What does that help mathematicians do?

Full Choice permits selecting one element from every set in any family of nonempty sets. Choice for ordinal-indexed families guarantees this for families indexed by ordinals, or well-ordered positions. The reported separation shows that even this restricted Choice combined with the Partition Principle cannot yield full Choice. Crucially, mapping partition parts injectively into the original set need not assign each part an element belonging to it.

Are there practical applications?

The immediate value is foundational: the reported models provide settings for distinguishing size-comparison principles from selection principles. The separate extension construction also preserves the original ordinals and adds no countable sequences of original elements. Researchers can therefore investigate failure of full Choice while keeping those specific features of the starting model unchanged.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Partition Principle does not imply Choice

September 24, 2026 44 pages Main result formalized in Lean

We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Partition-Principle-does-not-imply-Choice-September-24-2026,
  author = {{OpenAI}},
  title = {{The Partition Principle does not imply Choice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf}{OAI:The-Partition-Principle-does-not-imply-Choice-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/244.md.

The Partition Principle does not imply Choice

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Partition Principle does not imply Choice

Scope

The Partition Principle says that every surjection admits an injection in the reverse direction. The formalization proves the relative-consistency implication: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and failure of the Axiom of Choice.

The linked model-construction result also constructs a transitive model of the Partition Principle without Choice from an externally countable transitive ground model satisfying the stated axioms, including Choice, and containing an internal strongly inaccessible cardinal. The paper's stronger transitive-model preservation assertions are outside that selected construction.

Comparator links

Result	Comparator statement
Partition Principle without Choice from a ground model	PartitionPrinciple.lean
Relative consistency of the Partition Principle with ordinal Choice and failure of Choice	PartitionConsistency.lean