DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Choice for pairs and countable finite choice

Definition

Work in ZF. ${A C}_{2}$ says: for every set $I$ and family $(X_{i})_{i \in I}$ of two-element sets there is $c : I \to ⋃_{i \in I} X_{i}$ with $c (i) \in X_{i}$ for every $i$ . The restriction to $I = ω$ is ${A C}_{ω, 2}$ .

${A C}_{ω, f i n}$ says the same for every $ω$ -indexed family of nonempty finite sets. No ordering of the members of those finite sets is supplied. The empty index family has the empty choice function. These principles restrict the families in countable choice and AC; an arbitrary family of pairs need not be countably indexed.

Choice for pairs and countable finite choice

Definition

Depends on

Dependency tree · 2 levels

Sources