RemarkRemark: Literature-sourcedProof: Not suppliedSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

The Borel hierarchy on the real line never stabilizes at a countable stage

Statement

For an uncountable Polish space $X$ and every countable ordinal $1 \leq α < ω_{1}$ , Marker, Corollary 2.38, proves $Σ_{α}^{0} (X) \neq Π_{α}^{0} (X)$ and, in particular, $Σ_{α}^{0} (X) ⊊ Δ_{α + 1}^{0} (X)$ . Applied to $X = R$ , this says that alternating countable unions and countable intersections does not stabilize at any countable stage.

The strictness proof uses universal Borel sets and diagonalisation. Those descriptive-set-theoretic constructions are not developed here, so this result is recorded with its source rather than presented as a local theorem.

Used by

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

Dependency tree · next 3 levels

Nothing. This result depends on no other item in the library.

Sources

D. Marker, Descriptive Set Theory, Section 2, Corollary 2.38 (standard reference, not scraped)
M. Christ, Math 202B Lecture 1, Comment on the Borel hierarchy (standard reference, not scraped)