TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Sierpiński: arbitrary-set GCH implies AC in ZF

Statement

Over ZF, arbitrary-set GCH implies the Axiom of Choice.

Facts & Assumptions

[F1]

Local GCH for arbitrary sets: Arbitrary-set GCH supplies local GCH at every infinite set.

[F2]

Specker’s two-local-GCH theorem: A set containing omega and satisfying the two local hypotheses is well-orderable.

[F3]

The Axiom of Choice: A choice function selects a member of every set in a family of nonempty sets.

Proof

Given: The objects and hypotheses in the statement.

1.1

For any set $A$ , form $X = A ⊔ ω$ . The second summand explicitly embeds omega into $X$ , so $X$ and $P (X)$ are infinite. Global GCH supplies both local hypotheses. Specker well-orders $X$ , and restriction well-orders $A$ . This also covers finite and empty $A$ .

F1F2

2.1

For any family of nonempty sets, well-order its union by the preceding step and take the least member of each set. Replacement forms the resulting choice function. The empty family has the empty function. Thus AC holds.

F3step 1.1∎

Depends on

Used by

The aleph equation carries no well-orderability assertion False statement

Dependency tree · two levels

9 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

Caicedo, Some choiceless results (5), GCH consequence of Specker (standard reference, not scraped)
Carneiro, §2, local-to-global reduction, pp.1–2 (standard reference, not scraped)