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

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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.

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Vanishing at all positive exponents forces countability

Statement

Assume the Axiom of Countable Choice. The assertion “if $H^{s} (A) = 0$ for every finite $s > 0$ , then $A$ is countable” is false.

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, the square-position binary digit set is uncountable and has $H^{t} = 0$ for every finite $t > 0$ . An uncountable compact set can have dimension zero

Refutation

1.1

Take the square-position binary digit set $A_{S}$ . The cited result gives $H^{s} (A_{S}) = 0$ simultaneously for every finite $s > 0$ , as required by the antecedent.

2.1

That result also establishes that $A_{S}$ is uncountable, refuting the conclusion. The quantifier excludes exponent zero, so there is no assertion that its counting measure vanishes.

F1step 1.1∎

Depends on

An uncountable compact set can have dimension zero

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Bishop–Peres Example 1.4.2, square-position specialisation (standard reference, not scraped)