ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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 first constructible levels

Example

In ZF,

$L_{0} = \emptyset, L_{1} = {\emptyset}, L_{2} = {\emptyset, {\emptyset}}, L_{3} = P (L_{2}) .$

For every $n < ω$ , $L_{n} = V_{n}$ , and therefore $L_{ω} = V_{ω}$ .

Facts & Assumptions

Given: ZF. Calculated the first four levels and explicitly defined every subset of a finite carrier; finite induction and the omega union give the promised V-level identities.

[F1]

The constructible hierarchy and constructible rank: The L hierarchy starts from empty, uses Def at successors, and takes unions at nonzero limits.

[F2]

The cumulative hierarchy: The V hierarchy starts from empty, uses power sets at successors, and takes unions at nonzero limits.

[F3]

Definable subsets of a membership structure: Def includes each subset definable with finitely many parameters and has the designated empty case.

Verification

1.1F3

For a finite set A every subset B is finite. If $B = {b_{1}, \dots, b_{k}}$ is nonempty, the formula $x = b_{1} \lor \dots \lor x = b_{k}$ defines it over A using those parameters. The formula $x \neq x$ defines the empty subset when A is nonempty; the designated clause treats A empty. Hence $Def (A) = P (A)$ for finite A.

2.1F1F2step 1.1∎

Starting with empty gives $L_{1} = {\emptyset}$ and $L_{2} = {\emptyset, {\emptyset}}$ . The four elements of $L_{3}$ are $\emptyset$ , ${\emptyset}$ , ${{\emptyset}}$ and ${\emptyset, {\emptyset}}$ . Induction gives finiteness at each stage (a set with k elements has subsets encoded by k binary choices) and $L_{n + 1} = P (L_{n}) = P (V_{n}) = V_{n + 1}$ . The zero stages coincide. Taking the unions over n yields $L_{ω} = V_{ω}$ .

Depends on

Used by

Computing the canonical order at the first levels Example

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

Geschke Lemma 5.6(b) p15 (standard reference, not scraped)