DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Membership rank under Foundation

Definition

Now assume ZF, including Foundation. Membership on the universe is well-founded and setlike, so its ordinal rank is defined for every set. Write

$rank (x) = \sup {rank (y) + 1 : y \in x} .$

The empty supremum is $0$ , so $rank (\emptyset) = 0$ . If $y \in x$ , then $rank (y) < rank (x)$ . This is the Foundation-dependent special case of relation rank. The earlier construction of $V_{α}$ did not require Foundation.

Conventions and prerequisites: Ordinal rank of a well-founded relation, Equivalent forms of Foundation.

Depends on

Used by

V is a set False statement
A small transitive set bounds its ranks Lemma
Rank characterizes hierarchy membership Theorem

Dependency tree · two levels

7 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

Weiss, An Introduction to Set Theory (2014) — chapter 10 p.99. (standard reference, not scraped)