LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Every finite power of an at most countable set is at most countable

Statement

If $C$ is at most countable and $m\in\mathbb N$ , then the finite product $C^m=\prod_{i<m}C$ is at most countable.

Facts & Assumptions

Given: An at most countable set $C$ and a natural number $m$ .

[L1]

The product of two at most countable sets is at most countable (A product of two at most countable sets is at most countable).

[L2]

The product indexed by $0$ is a one-point set (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

[L3]

A finite set is at most countable, and induction holds on $\mathbb N$ (Finite, countably infinite, countable, uncountable, The principle of mathematical induction).

Proof

technique · induction

1.1

For $m=0$ , $C^0$ is a one-point set by [L2], hence at most countable by [L3].

baseL2L3

1.2

Assume $C^m$ is at most countable.

2.1

The product $C^{m+1}$ is canonically $C^m\times C$ , so it is at most countable by [L1].

step 1.2L1

3.1

By induction, $C^m$ is at most countable for every $m\in\mathbb N$ .

basestep 1.2step 2.1L3discharge-induction∎

Depends on

Used by

ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Countable set (standard reference, not scraped)