False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-09

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FALSE: every ω1-tree has a cofinal branch

Statement

Every $ω_{1}$ -tree has a cofinal branch, even when only normal splitting trees are considered.

Facts & Assumptions

Given: Work in ZFC. The statement above is to be refuted by the earlier constructed special Aronszajn tree.

[F1]

There exists a normal splitting special Aronszajn tree. A special Aronszajn tree exists

[F2]

A $κ$ -tree has height $κ$ and all levels of size less than $κ$ . κ-trees and the tree property

[F3]

An Aronszajn tree is an $ω_{1}$ -tree without a cofinal branch; a special tree admits a map to $ω$ injective on chains. Aronszajn, Suslin and special trees

[F4]

Under countable choice no countable subset of $ω_{1}$ is cofinal. Assuming countable choice: every at most countable subset of $ω_{1}$ is bounded below $ω_{1}$ , so no at most countable subset of $ω_{1}$ is cofinal in it, and a supremum of at most countably many at most countable ordinals is at most countable

[A1]

Assume AC, as required by the construction and F4. The Axiom of Choice

Refutation

1.1

Take the normal splitting special Aronszajn tree $T$ supplied by F1 under A1. By F3 and F2 it has height $ω_{1}$ and countable levels, so it satisfies the proposed hypothesis, including its optional normality and splitting restrictions. Specialness supplies $f : T \to ω$ with different values on comparable distinct nodes. In the construction this map is obtained by coding the strictly increasing rational labels by natural numbers.

F1F2F3A1given

2.1

If $B$ were a cofinal branch of $T$ , its nodes would be pairwise comparable, so $f ↾ B$ would be injective. Thus $B$ and its image under the height map would be countable. F4, using the countable choice supplied by A1, says that this height image cannot be cofinal in $ω_{1}$ . This contradicts the definition of a cofinal branch. Hence this concrete constructed tree refutes the statement.

F3F4A1step 1.1∎

Depends on

Used by

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Sources

Karagila, Axiomatic Set Theory, Theorem 9.2 and Exercise 9.4, printed p43; application of the local special-tree construction (standard reference, not scraped)