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 ultrafilter extension principle (UL/BPI)
Definition
The ultrafilter extension principle says that every filter on a set is contained in an ultrafilter on that set (Ultrafilter).
It is also called the ultrafilter lemma and is equivalent over ZF to the Boolean prime ideal theorem, abbreviated BPI. Items below write assume UL/BPI precisely when they use this extension principle; they do not thereby assume the full axiom of choice.
Depends on
Used by
- βℕ as the free ultrafilter algebra Example
- Under the ultrafilter lemma, an ultrafilter algebra maps each ultrafilter to its unique limit Lemma
- Under the ultrafilter lemma, closure in an ultrafilter-algebra topology is the image of ultrafilters containing the set Lemma
- Under the ultrafilter lemma, every ultrafilter algebra determines a compact Hausdorff topology Lemma
- Under the ultrafilter lemma, compact Hausdorff spaces and ultrafilter algebras are recovered by the two limit constructions Theorem
- Under the ultrafilter lemma, compact Hausdorff spaces are monadic over sets Theorem
Dependency tree · two levels
4 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
- J. Goubault-Larrecq, Algebras of filter-related monads: I. Ultrafilters and Manes' theorem (standard reference, not scraped)