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 set-theoretic cost of Hahn-Banach
Remark
The proof of Hahn-Banach dominated extension theorem for real vector spaces on this page is a Zorn proof, so that proof route uses the Axiom of Choice through Zorn's lemma. That is a proof cost, not the exact cost of the theorem itself.
The sharper ledger is recorded in The set-theoretic cost of Hahn-Banach ‡. For the reader of this page, the key points are these:
- the Boolean prime ideal theorem implies Hahn-Banach;
- relative to the consistency of ZF, Hahn-Banach does not imply the Boolean prime ideal theorem, so Hahn-Banach is strictly weaker than full choice by Halpern and Lévy 1971: the Boolean prime ideal theorem does not imply the Axiom of Choice ‡;
- Hahn-Banach already implies the existence of a non-Lebesgue measurable set and the Banach-Tarski paradox.
So later pages should cite Hahn-Banach itself when they use the extension theorem, and should cite Zorn only when they really use this maximal-extension implementation.
Depends on
Used by
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- W. A. J. Luxemburg, Two applications of the method of construction by ultrapowers to analysis (standard reference, not scraped)
- D. Pincus, The strength of the Hahn-Banach theorem (standard reference, not scraped)
- M. Foreman and F. Wehrung, The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set (standard reference, not scraped)
- J. Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox (standard reference, not scraped)