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The Partition Principle without Choice

Choice implies the Partition Principle, but an injection reversing a surjection need not select a member of each fiber.

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Understanding the problem

Read the background

Choice implies the Partition Principle, but an injection reversing a surjection need not select a member of each fiber. The unresolved target asks whether the cardinal comparison can recover full Choice.

This is a relative-consistency assertion under consistency of ZF. The companion transitive-model formulation has a stronger starting-model hypothesis; it is not an unconditional existence claim.

Why it matters

If correct, separates the Partition Principle from full Choice under the stated relative-consistency hypothesis.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • This is a relative-consistency assertion under consistency of ZF. The companion transitive-model formulation has a stronger starting-model hypothesis; it is not an unconditional existence claim.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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