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.

Ordinal rankings characterize well-foundedness

Statement

A definable setlike relation $R$ on $X$ is well-founded if and only if there is a definable ordinal-valued function $r$ on $X$ with $y R x \Rightarrow r (y) < r (x)$ . For well-founded $R$ , its rank $ρ_{R}$ is pointwise least among such functions. No choice or Foundation is needed.

Facts & Assumptions

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

[F1]

For a well-founded setlike relation $R$ on $X$ , its ordinal rank is the definable function determined by $ρ_{R} (x) = \sup {ρ_{R} (y) + 1 : y R x} = ⋃ {ρ_{R} (y) \cup {ρ_{R} (y)} : y R x} .$ To justify the definition, apply well-founded recursion to the total rule which returns this union if every value of its input function is an ordinal and returns $0$ otherwise. Well-founded induction shows that every actual value is an ordinal: predecessor values are ordinals by the induction hypothesis, their successors are ordinals, Replacement collects them, and their union is an ordinal, including the empty union $0$ . Thus the default case never occurs. For $y R x$ the rank equation gives $ρ_{R} (y) < ρ_{R} (x)$ . The definition requires no ambient Foundation for a supplied well-founded $R$ . Conventions and prerequisites: thm-recursion-on-well-founded-setlike-relations, lem-ordinal-basics. (Ordinal rank of a well-founded relation)

Proof

1.1

For well-founded $R$ , its recursively defined ordinal rank exists and strictly increases along each predecessor edge, providing a ranking.

1.2

Conversely, for a nonempty set $A \subseteq X$ , Replacement makes $r [A]$ a nonempty set of ordinals. It has a least element: choose one value $α$ and minimize within the set of values at most $α$ , using the well-order of $α + 1$ . A preimage $a \in A$ of that least value has no predecessor in $A$ , since a predecessor would have smaller rank.

given

2.1

Finally well-founded induction gives $ρ_{R} (x) \leq r (x)$ . If it holds at all $y R x$ , then $ρ_{R} (y) + 1 \leq r (y) + 1 \leq r (x)$ for all such $y$ , and taking the ordinal supremum gives the desired bound at $x$ . The empty supremum is zero.

F1step 1.2∎

Depends on

Ordinal rank of a well-founded relation

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

Marks, Set Theory, Berkeley edition — Exercise 6.8 p.31. (standard reference, not scraped)