Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Dimension zero forces countability

Statement

Assume the Axiom of Countable Choice. The assertion “every set of Hausdorff dimension zero 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 compact and uncountable with Hausdorff dimension zero. An uncountable compact set can have dimension zero

Refutation

1.1

Let S={k2:k≥1} and use the set AS of the cited counterexample. It has Hausdorff dimension zero.

F1
2.1

The same set is uncountable. Thus it satisfies the hypothesis but not the conclusion of the asserted implication.

F1step 1.1∎

Depends on

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