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

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BPI well-orders every set

False statement

The Boolean Prime Ideal Theorem implies that every set can be well-ordered.

Why this is false

Whenever the basic Cohen symmetric construction in F1 is supplied, its model satisfies BPI and contains an infinite Dedekind-finite set of reals, which cannot be well-ordered. Independently, F2 gives the exact syntactic nonimplication conditional on $Con (Z F)$ .

Facts & Assumptions

Given: Assume $Con (Z F)$ for the conditional nonimplication.

[F1]

The basic Cohen model satisfies BPI and fails Choice supplies the basic Cohen model and its infinite Dedekind-finite symmetric set $A$ .

[F2]

Relative consistency of BPI without Choice over ZF supplies the exact syntactic consistency implication from ZF to ZF+BPI+ $\neg$ AC.

[F3]

The Axiom of Choice defines AC as the assertion that every family of nonempty sets has a choice function.

Proof

technique · conditional countermodel

1.1

In the F1 model, suppose $A$ had a well-order. Recursively choose the least member not chosen earlier. If the recursion stopped, $A$ would be finite; if it did not, it would inject $ω$ into $A$ . Both alternatives contradict that $A$ is infinite and Dedekind-finite. Hence $A$ is not well-orderable although BPI holds.

F1assume-contra

2.1

Universal well-orderability implies F3 directly. Given a family $F$ of nonempty sets, well-order $⋃ F$ and assign to every $X \in F$ its least member; Replacement produces the resulting choice function. Therefore, if ZF+BPI proved universal well-orderability, it would prove AC. This contradicts the consistency of ZF+BPI+ $\neg$ AC supplied by F2 and gives the syntactic conditional counterexample.

F2F3step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)