Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
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.

The Solovay model fails the full Axiom of Choice

Statement

M does not satisfy full AC, though it satisfies DC.

Facts & Assumptions

Given: The internally proved ZF+DC theory of M.

[F1]

The Solovay model has no Vitali or Bernstein set: M has no Bernstein set.

[F2]

The Axiom of Choice and Choice gives a Bernstein set with no perfect-set, Baire or measure regularity: ZF+AC constructs a Bernstein subset of R.

Proof

1.1

Assume for contradiction that M⊨AC. Since M is a transitive ZF model with its own full real line, the proof in F2 relativizes to M and constructs there a Bernstein set.

assume-contraF2
2.1

This contradicts F1. Therefore M⊭AC. Its already proved DC is compatible with this failure because DC is strictly the serial omega-chain assertion, not a well-ordering principle.

discharge-contradiction: F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

27 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