Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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 ordered Mostowski relation has empty support

Statement

The dense order on the atoms is fixed by every allowed permutation and has empty support, although no well-order of the atoms is symmetric.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

The ordered Mostowski model defines the order-automorphism group and proves non-well-orderability.

Proof

1.1

For the relation R<={(a,b):a<b} and every allowed π, order preservation gives (a,b)∈R< iff (πa,πb)∈R<, hence πR<=R<. Its stabilizer is the whole group, so ∅ supports it.

F1
2.1

In contrast, if a well-order had finite support E, an order automorphism fixing E could move its least atom in A∖E, contradicting invariance. Thus “linearly ordered” and “well-orderable” separate in this model.

F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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