TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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Fodor’s pressing-down lemma

Statement

In ZFC, let $κ$ be regular uncountable, let $S \subseteq κ ∖ {0}$ be stationary, and let $f : S \to κ$ be regressive. Then some fibre ${α \in S : f (α) = ξ}$ is stationary.

Facts & Assumptions

[F1]

Regressive functions on ordinals: Regression means $f (α) < α$ throughout the nonzero domain.

[F2]

The club filter and nonstationary ideal: Nonstationary sets admit disjoint clubs, and stationary sets meet every club.

[F3]

The diagonal intersection of clubs is club: A kappa-indexed diagonal intersection of clubs is club.

Proof

Given: The objects and hypotheses in the statement.

1.1

If every fibre were nonstationary, ambient AC would select a club $C_{ξ}$ avoiding that fibre for every $ξ < κ$ . Let $D = △_{ξ < κ} C_{ξ}$ , a club.

F2F3

2.1

Take $α \in S \cap D$ . Then $α > 0$ and $ξ = f (α) < α$ , so diagonal membership gives $α \in C_{ξ}$ . But this club avoids the fibre containing $α$ , a contradiction. Therefore a stationary fibre exists.

F1F2step 1.1∎

Depends on

Used by

An unbounded regressive domain without a stationary fibre Counterexample
The transfinite subway argument Example
Splitting stationary sets of fixed smaller cofinality Lemma
Unboundedly many stationary fibres yield a partition Lemma
Solovay’s stationary partition theorem 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

Lietz, Lemma 5.11, pp.41–42 (standard reference, not scraped)
Vasey, Theorem 15.2, p.82 (standard reference, not scraped)