Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The exact choice strength of Stone's theorem remains open
Statement
As of 15 September 2026: AC proves Stone's theorem, while, assuming the consistency of ZF, neither DC nor BPI proves it; the stronger assertion that every open cover of every discrete metrizable space has a point-finite refinement equipped with a refinement map implies AC; the exact strength of the ordinary existential Stone theorem over is not identified here and remains open in the cited line of work.
Remarks
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The ZFC positive theorem. Stone's theorem, under choice: every metric space is paracompact proves that under AC every metric space is paracompact, and it is the positive half of the ledger.
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The two countermodels. Conditional on , Relative consistency of DC with failure of Stone's theorem gives a model of with a metrizable nonparacompact space, and Relative consistency of BPI with failure of Stone's theorem gives a model of with a metrizable nonmetacompact space (Metacompactness: every open cover has a point-finite open refinement); hence, under that same consistency hypothesis, neither DC nor BPI proves Stone's theorem.
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The stronger assertion. Effective metacompactness for discrete metric spaces implies AC proves that if every open cover of every discrete metrizable space has a point-finite refinement together with a refinement map, then the Axiom of Choice holds (The Axiom of Choice). The refinement map is part of the hypothesis: the statement is about a per-cover existential pair, not about a global class operator, and it is not identified with the ordinary Stone theorem.
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What remains open. The gap between the ordinary existential Stone theorem and the effective strengthening is not closed here: no model of is known to the cited authors in which Stone's theorem holds while the effective refinement assertion fails, and no proof of the ordinary theorem from a principle weaker than AC is known. The status is dated for that reason.
Depends on
- Relative consistency of DC with failure of Stone's theorem
- Relative consistency of BPI with failure of Stone's theorem
- Effective metacompactness for discrete metric spaces implies AC
- Stone's theorem, under choice: every metric space is paracompact
- Metacompactness: every open cover has a point-finite open refinement
- The Axiom of Choice
Used by
Dependency tree · two levels
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Sources
- C. Good, I. J. Tree, and W. S. Watson, On Stone's theorem and the axiom of choice (standard reference, not scraped)
- Samuel Corson, The Independence of Stone's Theorem from the Boolean Prime Ideal Theorem (standard reference, not scraped)