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.

The Lévy collapse of a regular uncountable cardinal

Statement

For regular uncountable $θ$ , $Lv (θ)$ makes every $α < θ$ countable while preserving $θ$ , so $θ$ is exactly the new $ℵ_{1}$ ; the example displays the dense sets making each coordinate map $ω$ onto $α$ .

Facts & Assumptions

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

[F1]

Cardinal effects of collapse and Lévy-collapse forcing proves the general effect and preservation statement.

Proof

1.1

Fix $0 < α < θ$ . For $n \in ω$ let $D_{n} = {p : (α, n) \in dom p}$ , and for $β < α$ let $E_{β} = {p : \exists n p (α, n) = β}$ . Extend a finite condition at the requested coordinate, choosing a fresh $n$ for $E_{β}$ , so both families are dense. A generic meets them all, and $g_{α} (n) = (⋃ G) (α, n)$ is a surjection $ω ↠ α$ .

2.1

F1 gives $θ$ -cc, hence preserves the cardinal $θ$ . Since every infinite ordinal below it is countable by step 1.1, $θ$ is the least uncountable cardinal in the extension, namely $ℵ_{1}$ .

F1step 1.1∎

Depends on

Cardinal effects of collapse and Lévy-collapse forcing

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, Lévy collapse (standard reference, not scraped)