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.
This establishes AC (The Axiom of Choice) as a sufficient hypothesis for the extension theorem proved here. It establishes no converse implication or strict comparison with another choice principle. The exact use of Zorn is to select a maximal dominated extension; the one-step lemma then forces its domain to be the whole vector space.
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
11 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)