Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17‡ rests on unproved material
How 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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: every Borel subset of the real line is a countable union of countable intersections of open and closed sets

Statement

Every Borel subset of R belongs to the class obtained by taking countable intersections of open and closed sets and then countable unions of those intersections.

Facts & Assumptions

Given: The Borel hierarchy on R formed by alternating countable unions and countable intersections from the open and closed sets.

‡ [L1]

For every ordinal 1≤α<ω1, the additive and multiplicative Borel classes on R differ, so the Borel hierarchy does not stabilize at any countable stage (The Borel hierarchy on the real line never stabilizes at a countable stage ‡).

Refutation

technique · contradiction
1.1

Suppose, for contradiction, that every Borel set belongs to the displayed fixed finite-stage class.

assume-contra
2.1

That class would then equal the Borel sigma-algebra. Since the Borel sigma-algebra is already closed under complements, countable unions, and countable intersections, every further stage would add no set, so the hierarchy would stabilize at that countable stage.

step 1.1algebra
3.1

The stabilization in step 2.1 contradicts [L1]. Hence the asserted finite description does not contain every Borel subset of R.

step 2.1‡ L1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 1 result over 1 level. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources