Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26
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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.

FALSE: assuming the Axiom of Choice, every subset of R is Lebesgue measurable

Statement

Assume the Axiom of Choice. Every subset of R is Lebesgue measurable.

Facts & Assumptions

Given: The Axiom of Choice.

[L1]

Assuming the Axiom of Choice, a Vitali set in [0,1] exists (Assuming choice on the cosets of Q in R, a Vitali set in [0,1] exists).

[L2]

Assuming the Axiom of Choice, every Vitali set is not Lebesgue measurable (Assuming the Axiom of Choice, a Vitali set is not Lebesgue measurable).

Refutation

technique · direct
1.1

By [L1] choose a Vitali set V⊆[0,1].

L1
2.1

The set V is a subset of R and is not Lebesgue measurable by [L2], so the universal claim is false.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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