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 Boolean prime ideal principle
Definition
The Boolean prime ideal principle (BPI) asserts that every nontrivial Boolean algebra has a prime ideal. Properness is part of primality.
The set ultrafilter lemma (UFL) asserts that for every set , every proper filter of subsets of extends to a maximal proper filter of subsets of . Here a set filter contains , excludes , is upward closed within and is closed under finite intersections.
These are principles considered over ZF. Naming either principle is not assuming it or asserting a ZF proof of it. On there is no proper set filter, so the corresponding UFL instance is vacuous. BPI excludes the trivial algebra from its existence assertion.
Depends on
Used by
Dependency tree · two levels
2 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
- Tressl, Stone Duality for Boolean Algebras, 2.2.11 and 2.3.3 (existence and prime-filter conventions) (standard reference, not scraped)