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.

Local GCH for arbitrary sets

Definition

Write $X ⪯ Y$ for an injection and $X \approx Y$ for a bijection; $X ≺ Y$ means $X ⪯ Y$ and $X \approx̸ Y$ . For an infinite set $X$ , local GCH at $X$ , denoted $C H (X)$ , means

$\forall Y (X ⪯ Y ⪯ P (X) ⟹ Y \approx X or Y \approx P (X)) .$

Arbitrary-set GCH asserts this for every infinite set. No well-order of $X$ is implicit. We write $Seq (X) = ⋃_{n < ω}^{n} X$ for finite sequences, including the empty sequence, and $h (X)$ for the Hartogs number. Disjoint union is denoted $⊔$ .

Depends on

Used by

The aleph equation carries no well-orderability assertion False statement
Local GCH absorbs sums and squares Lemma
Sierpiński: arbitrary-set GCH implies AC in ZF Theorem

Dependency tree · two levels

3 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

Carneiro, GCH implies AC, §§1–2, pp.1–2 (standard reference, not scraped)
Caicedo, Some choiceless results (5), opening Specker theorem (standard reference, not scraped)