CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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Finite strictly decreasing sequences of naturals form a tree with every finite level nonempty but no infinite branch

Statement refuted

In König's infinity lemma: an ordered finitely branching tree with a node at every level has an infinite branch, in ZF, finite branching can be replaced by arbitrary branching: every tree of finite sequences with a node at every level has an infinite branch.

Facts & Assumptions

Given: The tree conventions of Rooted trees of finite sequences, levels, branches, and finite branching, with ordered finite successor sets.

[L1]

An ordered finitely branching tree with a node at every level has an infinite branch, in ZF (König's infinity lemma: an ordered finitely branching tree with a node at every level has an infinite branch, in ZF).

Counterexample

technique · constructive

1.1

Let $T$ consist of the empty sequence and all finite strictly decreasing sequences of natural numbers. It is prefix closed. For every $n$ , the sequence $(n - 1, n - 2, \dots, 0)$ is a node of length $n$ , so every finite level is nonempty.

construct

2.1

The root has infinitely many successors, so $T$ is not finitely branching. An infinite branch would be an infinite strictly decreasing sequence of naturals, but its range would have a least element by The well-ordering principle, after which the branch would have to contain a smaller one. Thus no infinite branch exists, and the missing hypothesis relative to [L1] is exactly finite branching.

step 1.1L1discharge-construct∎

Depends on

Used by

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Dependency tree · two levels

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