LemmaStatement: Literature-sourcedProof: AI-generatedprecheck 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 $m \in N$ , 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 $N$ (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 $m \in N$ .

basestep 1.2step 2.1L3discharge-induction∎

Depends on

Used by

Dependency tree · two levels

22 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

Countable set (standard reference, not scraped)