Alphabeta Math
CorollaryStatement: 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.

Externally fixed-fragment relative consistency of not CH and not GCH

Statement

Externally, Con⁡(ZFC) implies each of Con⁡(ZFC+¬CH) and Con⁡(ZFC+¬GCH). The implication here is a metatheorem obtained by applying a separate finite-fragment construction to any purported contradiction proof; no PA proof of a uniform refutation transformer is claimed.

Facts & Assumptions

Given: The ordinary syntactic consistency predicate and one hypothetical finite refutation.

[F1]

Fixed finite-fragment verification for the Cohen countermodels gives, for each externally fixed finite fragment of either target theory, a ZFC proof that a model of that fragment exists.

Proof

1.1

If ZFC+¬CH were inconsistent, a contradiction proof would use only a finite set Δ of its axioms. Fix that Δ externally. F1 gives a ZFC proof of the existence of a model of Δ. Soundness for the finite proof shows that no such model exists, so ZFC itself would be inconsistent. Contraposition gives the first consistency implication.

F1
2.1

The same argument with a finite refutation of ZFC+¬GCH and the second F1 construction gives the other implication. Both are external consequences of fixed-fragment proofs; the positive consistency counterparts are separate results.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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