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.

Dimensions add under unions

Statement

Assume the Axiom of Countable Choice. The assertion dim⁡H(A∪B)=dim⁡HA+dim⁡HB for all subsets of a metric space is false, even for disjoint compact subsets of the line.

Facts & Assumptions

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

[F1]

Under the standing Countable Choice hypothesis, the dimension of a countable union is the supremum of the component dimensions. Hausdorff dimension is monotone and countably stable

[F2]

Under the standing Countable Choice hypothesis, a positive-length subset of the line has dimension one. Euclidean space and positive-volume sets have their Euclidean dimension

Refutation

1.1

Let A=[0,1] and B=[2,3]. They are disjoint compact intervals, each with positive length, so dim⁡HA=dim⁡HB=1.

F2
2.1

Countable stability applied to these two sets and empty remaining terms gives dim⁡H(A∪B)=max⁡(1,1)=1. This differs from 1+1=2, refuting the asserted sum rule.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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