Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

First hierarchy stages and their ranks

Example

The first hierarchy stages are V0=∅, V1={∅} and V2={∅,{∅}}. Their ranks are respectively 0,1,2. Also rank⁡({{∅}})=2, even though this last singleton is not the ordinal 2.

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

In ZF, for every ordinal α, rank⁡(α)=α and rank⁡(Vα)=α. (Ranks of ordinals and hierarchy stages)

Verification

1.1

Unfolding the empty stage and two power sets gives the displayed sets. The rank-of-stages formula gives rank⁡(Vi)=i for i=0,1,2.

F1
2.1

The membership-rank equation gives rank⁡({{∅}})=rank⁡({∅})+1=2. This singleton omits ∅, while the ordinal 2={∅,{∅}} contains it. Equal ranks therefore do not identify sets.

F1step 1.1∎

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