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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
- A consistent BPI counterexample theory obstructs a proof of DMC Corollary
- The tail-flip symmetric model refutes BPI Corollary
- BPI is equivalent to the Axiom of Choice False statement
- Choice and forcing boundary Remark
- AC implies BPI Theorem
- BPI and the set ultrafilter lemma are equivalent over ZF Theorem
- BPI is equivalent to extending proper Boolean filters Theorem
- Extreme amenability yields BPI in finite-support permutation models Theorem
- Products of cofinite spaces are compact exactly under BPI Theorem
- The basic Cohen model satisfies BPI and fails Choice Theorem
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)