Alphabeta MathHow 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.
The rationals are dense but have dimension zero
Example
Assume the Axiom of Countable Choice. The set has , although it is dense in and its closure has dimension one. Hausdorff dimension need not be preserved by taking closure.
Facts & Assumptions
Given: The objects, conventions, and hypotheses in the statement above.
Under the standing Countable Choice hypothesis, at most countable sets have Hausdorff dimension zero. Hausdorff dimension is monotone and countably stable
Under the standing Countable Choice hypothesis, a subset of with positive Lebesgue outer measure has dimension one. Euclidean space and positive-volume sets have their Euclidean dimension
The embedded rationals are dense in the real line. Both and are dense in , and every nonempty open subset of is uncountable
The rationals are countably infinite. is countably infinite
Verification
As a subset of the countable rationals, is at most countable; hence . The endpoints zero and one are included.
Every relative neighbourhood in contains a rational point of , by density (and the endpoints themselves at the ends). Thus , whose Lebesgue measure is one and whose dimension is consequently one.
Depends on
- Hausdorff dimension is monotone and countably stable
- Euclidean space and positive-volume sets have their Euclidean dimension
- Both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable
- $\mathbb{Q}$ is countably infinite
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
32 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 1.2.7 (countable nullity), specialised to Q (standard reference, not scraped)