PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Basic stationary-set calculus

Statement

In ZFC, for $cf (θ) > ω$ : stationary subsets of $θ$ are unbounded; every club is stationary; supersets of stationary sets are stationary; the intersection of a stationary set with a club is stationary; and a union of fewer than $cf (θ)$ nonstationary sets is nonstationary.

Facts & Assumptions

[F1]

The club filter and nonstationary ideal: Stationarity means meeting every club; the club filter and its dual ideal are closed under the stated small intersections and unions.

Proof

Given: The objects and hypotheses in the statement.

1.1

Every tail $[β, θ)$ is closed and unbounded: for any bound take a larger ordinal above $β$ , and a limit of tail points is still at least $β$ . A bounded set is disjoint from a suitable tail, so cannot be stationary. This also excludes the empty set and all singletons.

1.2

Two clubs intersect in a club and hence nontrivially, so each club is stationary. Supersets preserve intersections with every club. For stationary $S$ and clubs $C, D$ , the club $C \cap D$ meets $S$ , so $S \cap C$ meets every $D$ and is stationary.

2.1

For a small family of nonstationary sets, their union is in the dual ideal by its completeness. Explicitly choose an avoiding club for each member and intersect those clubs; the resulting club avoids the union. For the empty family the union is empty, avoided by $θ$ .

F1∎

Depends on

The club filter and nonstationary ideal

Used by

The transfinite subway argument Example
Removing the trace preserves a stationary remainder Lemma
Unboundedly many stationary fibres yield a partition Lemma
Fodor’s pressing-down lemma Theorem
Solovay’s stationary partition theorem Theorem

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

Vasey, Example 14.13(1)–(5), p.81 (standard reference, not scraped)
Lietz, Lemma 5.6 and Definition 5.7 (standard reference, not scraped)