False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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BPI is equivalent to the Axiom of Choice

False statement

The Boolean Prime Ideal Theorem is equivalent over ZF to the Axiom of Choice.

Why this is false

Conditional on $Con (Z F)$ , AC strictly implies BPI over ZF: AC proves BPI, while BPI does not prove AC.

Facts & Assumptions

Given: Work over ZF. For the strictness assertion assume $Con (Z F)$ .

[F1]

The Boolean prime ideal principle and The Axiom of Choice state BPI and AC.

[F2]

Relative consistency of BPI without Choice over ZF gives the exact syntactic implication $Con (Z F) \Rightarrow Con (Z F + B P I + \neg A C)$ .

[F3]

AC implies BPI proves in ZF that AC implies BPI, with the exact Zorn argument and degenerate Boolean-algebra case.

Proof

technique · implication plus conditional countermodel

1.1

F3 gives $A C \Rightarrow B P I$ over ZF.

F1F3

1.2

If ZF+BPI proved AC, then adding $\neg$ AC would make ZF+BPI+ $\neg$ AC inconsistent. Under the given consistency hypothesis this contradicts F2.

F2assume-contra

2.1

Thus, conditional on $Con (Z F)$ , the reverse implication fails while the forward implication of step 1.1 holds. The false equivalence is refuted with exactly the stated consistency qualification.

step 1.1step 1.2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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. D. Halpern and A. Lévy, The Boolean prime ideal theorem does not imply the axiom of choice, pp.83-134 (standard reference, not scraped)