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.

An orbit set can be symmetric when its enumeration is not

Statement

In the basic Cohen system, $A = {a_{n} : n \in ω}$ has empty support, although the graph $e = {⟨ n, a_{n} ⟩ : n \in ω}$ is moved by a transposition outside every finite support. More generally, no enumeration of $A$ belongs to the symmetric model, as the supplier's all-enumerations argument proves.

Facts & Assumptions

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

[F1]

The Cohen reals form a symmetric set but their enumeration is not symmetric gives $π a_{n} = a_{π (n)}$ , proves that the $a_{n}$ are pairwise distinct, and computes the supports of $A$ and its canonical enumeration.

Proof

1.1

For every finite permutation $π$ , $π A = {π a_{n} : n \in ω} = {a_{π (n)} : n \in ω} = A .$ Thus the whole automorphism group stabilizes $A$ , so $\emptyset$ is a support.

1.2

Let $E \subseteq ω$ be finite. Choose distinct $n, m \notin E$ and let $π = (n m)$ . Then $π \in fix (E)$ , but the pair $⟨ n, a_{n} ⟩$ is sent to $⟨ n, a_{m} ⟩$ in the action on the graph's value coordinate after the ground ordinal $n$ is fixed. Since $a_{n} \neq a_{m}$ , $π e \neq e$ . Hence no finite $E$ supports $e$ .

2.1

The calculation separates an invariant unordered range from a non-symmetric ordering of that range. It does not claim merely that this particular graph is absent: F1 separately supplies the support argument excluding every enumeration. No Choice is used.

F1step 1.1step 1.2∎

Depends on

The Cohen reals form a symmetric set but their enumeration is not symmetric

Used by

Nothing in the library uses this result yet.

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

Karagila, Forcing & Symmetric Extensions, Propositions 10.22–10.24, p. 50 (standard reference, not scraped)