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.
Martin's Axiom at a cardinal and Martin's Axiom
Definition
Work in ZFC. For an infinite cardinal , says that whenever is a nonempty ccc forcing partial order and is a family of at most dense subsets of , there is a filter meeting every member of . Here filters use the stronger-is-smaller convention. Replacing each by its downward closure shows that dense and dense-open formulations agree.
Martin's Axiom, MA, is the scheme for every infinite . The inequality is strict; is not included.
Depends on
Used by
- MA produces a real outside a small listed family Example
- Martin's Axiom reduces to small ccc orders Lemma
- Cardinal exponentiation below the continuum under MA Theorem
- MA makes unions of fewer than continuum many meagre sets meagre Theorem
- MA makes unions of fewer than continuum many null sets null Theorem
- MA(aleph₀) and the implication from CH to MA Theorem
- Under MA(aleph₁), arbitrary products of ccc spaces are ccc Theorem
Dependency tree · two levels
22 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
- Karagila, Forcing & Symmetric Extensions, Definition 7.1 (standard reference, not scraped)