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.

Specker’s two-local-GCH theorem

Statement

In ZF, if $ω ⪯ X$ , $C H (X)$ and $C H (P (X))$ , then $P (X) \approx h (X)$ . In particular $X$ is well-orderable.

Facts & Assumptions

[F1]

The local-GCH Hartogs dichotomy: For a set containing omega and satisfying local GCH, an embedding of its Hartogs number into its power set gives that power set equinumerous with the Hartogs number; otherwise the Hartogs numbers agree.

[F2]

Hartogs bounds in iterated power sets: If $X^{2} \approx X$ , then $h (X) ⪯ P^{2} (X)$ .

[F3]

Local GCH absorbs sums and squares: The local hypotheses imply $X^{2} \approx X$ .

Proof

Given: The objects and hypotheses in the statement.

1.1

Write $P = P (X)$ and $Q = P (P)$ . Suppose $P$ is not well-orderable. Then $Q$ is not well-orderable either, since $P ⪯ Q$ . Apply the dichotomy to $X$ and to $P$ ; both contain an injected omega and satisfy their respective local hypotheses. Their first branches are excluded, so $h (X) = h (P) = h (Q)$ .

2.1

But square absorption and the double-power bound give $h (X) ⪯ Q$ . Together with the equality in the previous step this embeds the Hartogs number of $Q$ into $Q$ , impossible by its defining property. Thus $P$ is well-orderable.

F2F3step 1.1

3.1

A well-order of $P$ has some ordinal type $ρ$ . If $ρ < h (X)$ , it would embed into $X$ , giving $P ⪯ X$ , which is impossible: invert that injection on its range and extend by the empty subset elsewhere to get a surjection $s : X \to P (X)$ ; then ${x \in X : x \notin s (x)}$ is missed. Thus $h (X) \leq ρ$ and $h (X) ⪯ P$ . The first dichotomy branch now gives $P \approx h (X)$ and the well-orderability of $X$ .

F1step 2.1∎

Depends on

Used by

The two local hypotheses at omega Example
Sierpiński: arbitrary-set GCH implies AC in ZF Theorem

Dependency tree · two levels

12 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), Specker theorem and complete proof (standard reference, not scraped)
Carneiro, Theorem 1, p.2 (standard reference, not scraped)