DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Closed unbounded subsets of ordinals

Definition

For a nonzero limit ordinal $θ$ , a subset $C \subseteq θ$ is unbounded if $\forall β < θ \exists γ \in C (β < γ)$ . Put

${acc}_{θ} (C) = {δ < θ : δ is a nonzero limit ordinal and \sup (C \cap δ) = δ} .$

It is closed if ${acc}_{θ} (C) \subseteq C$ , and club if closed and unbounded. Neither $0$ nor $θ$ is required to belong to a club. We omit the subscript on acc when the ambient ordinal is clear. The main setting on this page is ZFC and a regular uncountable cardinal $κ$ ; some lemmas explicitly allow $θ$ with merely uncountable cofinality.

Depends on

Cofinality $\operatorname{cf}(\alpha)$, and regular and singular cardinals

Used by

Diagonal intersection and union Definition
Normal ordinal functions Definition
Tails, limits, and diagonal intersection Example
Countable intersections of clubs are always club False statement
Closure points form a club Lemma
Intersections of fewer than the cofinality many clubs 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, §5.1 Definition 5.1, printed p.39 (PDF p.40) (standard reference, not scraped)
Vasey, §14 Definition 14.1, p.79 (standard reference, not scraped)