False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Countable intersections of clubs are always club

Statement

False without a cofinality restriction: for every limit ordinal $θ$ , a countable intersection of clubs in $θ$ is club.

Facts & Assumptions

[F1]

Closed unbounded subsets of ordinals: Club means closed and unbounded in the ambient ordinal.

[F2]

Intersections of fewer than the cofinality many clubs: The intersection theorem requires the indexing size strictly below the ambient cofinality, with uncountable ambient cofinality.

Refutation

Given: The objects and hypotheses in the statement.

1.1

At theta equal to omega, each tail $C_{n} = {m < ω : n \leq m}$ is unbounded and vacuously closed, because omega has no nonzero limit ordinal below it. Their intersection is empty and hence not club.

2.1

Even at the uncountable singular ordinal $θ = ℵ_{ω}$ , the tails $D_{n} = [ℵ_{n}, ℵ_{ω})$ are club, but their intersection is empty since the alephs indexed by natural numbers are cofinal in theta. In both cases the countable index size equals, rather than lies strictly below, the ambient cofinality omega. Thus neither example meets the intersection theorem's hypotheses.

F1F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, Definition 5.7 and following completeness argument, pp.40–41 (standard reference, not scraped)
Vasey, Fact 14.8, pp.80–81 (standard reference, not scraped)