PropositionStatement: AI-adaptedProof: 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.

Ranks of ordinals and hierarchy stages

Statement

In ZF, for every ordinal $α$ , $rank (α) = α$ and $rank (V_{α}) = α$ .

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 $α$ , $x \in V_{α} ⟺ rank (x) < α, x \subseteq V_{α} ⟺ rank (x) \leq α .$ Thus $rank (x)$ is the least $α$ with $x \subseteq V_{α}$ , and $rank (x) = α$ iff $x \in V_{α + 1} ∖ V_{α}$ . (Rank characterizes hierarchy membership)

[F2]

In ZF without Foundation, every $V_{α}$ is transitive and $α \leq β$ implies $V_{α} \subseteq V_{β}$ . Also $V_{α} \cap Ord = α$ , and both $α$ and $V_{α}$ belong to $V_{α + 1} ∖ V_{α}$ . (Transitivity and growth of hierarchy stages)

Proof

1.1F1F2

By F2, $α \in V_{α + 1} ∖ V_{α}$ . The final equivalence in F1 therefore gives $rank (α) = α$ , including $α = 0$ .

2.1F1F2∎

By F2, $V_{α} \in V_{α + 1} ∖ V_{α}$ . Applying the same equivalence in F1 gives $rank (V_{α}) = α$ , including $V_{0} = \emptyset$ .

Depends on

Used by

A singleton can have large rank and hereditary size Example
First hierarchy stages and their ranks Example
Grothendieck universes and relative size Remark

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

Marks, Set Theory, Berkeley edition — 7.4 p.34. (standard reference, not scraped)