DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

κ-trees and the tree property

Definition

Let $κ$ be an infinite cardinal in the sense of Cardinal (initial ordinal) and cardinality. A $κ$ -tree is a tree $T$ of height $κ$ such that $∣ T_{α} ∣ < κ$ for every $α < κ$ , with height and levels as in Set-theoretic trees, heights, levels, branches and antichains. The tree property at $κ$ asserts that every $κ$ -tree has a cofinal branch.

Regularity is a separate condition ( $cf (κ) = κ$ , using Cofinality $cf (α)$ , and regular and singular cardinals); it is not imposed by this definition. In particular an $ω$ -tree has height $ω$ and finite levels. Having $κ$ nodes alone is not the definition of a $κ$ -tree. Empty and singleton trees have heights zero and one, respectively, so are not $κ$ -trees for the permitted infinite cardinals.

Depends on

Used by

Aronszajn, Suslin and special trees Definition
FALSE: every ω1-tree has a cofinal branch False statement
Splitting turns an uncountable branch into an antichain Lemma

Dependency tree · two levels

18 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

Monk, Set theory following Jech (2024), Chapter 9, printed p65; cardinal-tree convention made explicit (standard reference, not scraped)