Alphabeta Math
CorollaryStatement: Literature-sourcedProof: 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.

The universe is the class union of its stages

Statement

In ZF every set x belongs to Vrank⁡(x)+1. Consequently V=⋃α∈Ord⁡Vα in the class sense: every set lies in a stage. This is not a union indexed by a set of all ordinals.

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 set x and ordinal α, x∈Vα  ⟺  rank⁡(x)<α,x⊆Vα  ⟺  rank⁡(x)≤α. Thus rank⁡(x) is the least α with x⊆Vα, and rank⁡(x)=α iff x∈Vα+1∖Vα. (Rank characterizes hierarchy membership)

Proof

1.1F1

For ρ=rank⁡(x), the ordinal inequality ρ<ρ+1 and the membership characterization give x∈Vρ+1.

2.1step 1.1∎

Every member of a stage is a set, and step 1.1 gives a stage containing every set. Thus the two class descriptions agree. Neither statement asserts that their collection is a set.

Depends on

Used by

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