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.

Multiple choice and dependent multiple choice

Definition

Work in ZF. Multiple choice (MC) asserts that every set-indexed family $(X_{i})_{i \in I}$ of nonempty sets admits a function $i \mapsto F_{i}$ with $\emptyset \neq F_{i} \subseteq X_{i}$ finite. Countable multiple choice (CMC) restricts this to $I = ω$ .

Dependent multiple choice (DMC) asserts: if $X \neq \emptyset$ and $R \subseteq X^{2}$ satisfies $\forall x \in X \exists y \in X (x R y)$ , there is a sequence $(F_{n})_{n < ω}$ of nonempty finite subsets of $X$ such that

$\forall n < ω \forall x \in F_{n} \exists y \in F_{n + 1} (x R y) .$

No initial $F_{0}$ is prescribed. The condition requires a successor for every point at each level. It does not require every point of the next level to have a predecessor.

Depends on

Used by

Multiple choice produces maximal antichains Lemma
DC and finite multiple selections Theorem

Dependency tree · two levels

7 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

Jech, The Axiom of Choice, §9.1, p.133 (standard reference, not scraped)
alg-d, On dependent choice, DMC definition and Proposition 6, PDF p.4 (standard reference, not scraped)