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.

Relative consistency of ZF with failure of Choice

Statement

Externally, Con⁡(ZF) implies Con⁡(ZF+¬AC). The implication uses separately fixed finite-fragment model proofs; no PA-verified uniform symmetric-model proof transformer or transitive model of full ZF is asserted.

Facts & Assumptions

Given: One hypothetical finite contradiction proof from ZF+¬AC.

[F1]

Formal consistency of ZFC plus GCH relative to ZF gives Con⁡(ZF)⇒Con⁡(ZFC+GCH), and hence consistency of ZFC.

[F2]

Fixed finite-fragment verification for the basic Cohen symmetric model gives a ZFC proof of a set model for every externally fixed finite target fragment.

[F3]

The Axiom of Choice is the sentence negated in the target.

Proof

1.1F2F3

The hypothetical contradiction proof uses a finite list Δ of axioms of ZF and the negation of F3. Fix this list externally. F2 provides a ZFC proof that a set model of Δ exists. Soundness for the particular finite contradiction proof gives a ZFC proof that no such model exists. Thus target inconsistency implies inconsistency of ZFC.

2.1F1step 1.1∎

F1 makes ZFC consistent whenever ZF is consistent. Contraposition in step 1.1 therefore proves the stated external relative-consistency implication. Source Choice is available in the ambient ZFC construction; the symmetric target refutes it. Each target proof is handled separately, without a claim that PA verifies a uniform map on proof codes.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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