Alphabeta MathHow 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, implies each of and . 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.
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
If were inconsistent, a contradiction proof would use only a finite set of its axioms. Fix that externally. F1 gives a ZFC proof of the existence of a model of . Soundness for the finite proof shows that no such model exists, so ZFC itself would be inconsistent. Contraposition gives the first consistency implication.
The same argument with a finite refutation of and the second F1 construction gives the other implication. Both are external consequences of fixed-fragment proofs; the positive consistency counterparts are separate results.