ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

MA produces a real outside a small listed family

Statement

Assume $M A (κ)$ . Given at most $κ$ reals, the Cohen finite-function order and coordinate-domain/disagreement dense sets produce a real distinct from every listed real. Hence $M A (κ)$ implies $κ < 2^{ℵ_{0}}$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Martin's Axiom at a cardinal and Martin's Axiom supplies a filter meeting the dense family.

[F2]

Cohen, collapse, and Lévy-collapse forcing orders defines $Add (ω, 1)$ .

Proof

1.1

List the given reals as $r_{α}$ for $α < λ \leq κ$ . In $P = Add (ω, 1)$ let $E_{n} = {p : (0, n) \in dom p}$ and $D_{α} = {p : \exists n ((0, n) \in dom p \land p (0, n) \neq r_{α} (n))}$ . Extending at one fresh coordinate proves all these sets dense; $P$ is countable and hence ccc.

2.1

F1 supplies a filter meeting the at most $κ$ many $D_{α}$ and countably many $E_{n}$ . Its union is a total real $g$ , and meeting $D_{α}$ gives $g \neq r_{α}$ . Thus no family of at most $κ$ reals exhausts $2^{ω}$ , so $κ < 2^{ℵ_{0}}$ .

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Karagila, Forcing & Symmetric Extensions, Proposition 7.5 (standard reference, not scraped)