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 MA and not CH

Statement

Externally, Con⁡(ZFC) implies Con⁡(ZFC+MA+¬CH). This is a fixed finite-fragment metatheorem, without a claim that PA verifies a uniform proof-code reduction.

Facts & Assumptions

Given: One hypothetical finite contradiction proof in the target theory.

[F1]

Fixed finite-fragment verification for the MA iteration gives a ZFC proof of a model of every externally fixed finite target fragment.

Proof

1.1

A contradiction proof in ZFC+MA+¬CH uses some finite set Δ of target axioms, including finitely many MA instances. Apply F1 to that particular Δ. ZFC proves that a model of Δ exists, while the alleged contradiction proof and finite-model soundness give a ZFC proof that no such model exists. Thus a target inconsistency would imply a ZFC inconsistency.

F1
2.1

Contraposition yields the displayed external relative-consistency implication. No assertion about PA verification of the fragment-selection map is needed.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

5 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