CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Extensionality is needed for injective collapse

Statement refuted

False claim: every well-founded relation has an injective collapse. Let $X = {a, b}$ with $a \neq b$ and $R = \emptyset$ .

Facts & Assumptions

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

[F1]

A setlike relation $R$ on $X$ is extensional when ${pred}_{R} (x) = {pred}_{R} (y)$ implies $x = y$ for $x, y \in X$ . If $R$ is also well-founded, its collapse map is the unique definable function $π (x) = {π (y) : y R x} .$ Existence and uniqueness follow from well-founded recursion with $G (x, h) = ran (h)$ , a set by Replacement. A collapse map is defined even without extensionality; injectivity is a further conclusion requiring extensionality. The construction for a supplied well-founded relation uses no ambient Foundation. Conventions and prerequisites: thm-recursion-on-well-founded-setlike-relations. (Extensional relations and collapse maps)

Counterexample

1.1

Every member of every nonempty subset of $X$ is minimal, so $R$ is well-founded and setlike. Its two predecessor sets are both empty, so it is not extensional.

2.1

The collapse rule gives $π (a) = \emptyset = π (b)$ , because both predecessor images are empty. Thus the collapse exists but is not injective.

F1step 1.1∎

Depends on

Extensional relations and collapse maps

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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 — 6.10–6.11 pp.31–32, sharpness instance. (standard reference, not scraped)