LemmaStatement: Literature-sourcedProof: 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.

Canonical check names are hereditarily symmetric

Statement

Every ground set $x$ has a check name fixed by every forcing automorphism; it is hereditarily symmetric and evaluates to $x$ . Thus the ground model lies in every symmetric extension.

Facts & Assumptions

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

[F1]

Automorphisms acting on forcing names gives the recursive action and check-name fixation.

[F2]

Symmetric forcing systems, supports, and hereditarily symmetric names defines HS.

[F3]

Check-name evaluation and reconstruction of G gives ${\overset{ˇ}{x}}_{G} = x$ .

Proof

1.1F1F2

Induct on rank of $x$ . Every subname $\overset{ˇ}{y}$ for $y \in x$ is HS by induction. F1 gives $π \overset{ˇ}{x} = \overset{ˇ}{x}$ for every automorphism, so its stabilizer is the whole group and belongs to the normal filter. Thus $\overset{ˇ}{x}$ is HS.

2.1F3step 1.1∎

F3 gives ${\overset{ˇ}{x}}_{G_{0}} = x$ . Hence every ground set occurs as the value of an HS name, including $x = \emptyset$ , and the ground model is contained in the symmetric interpretation.

Depends on

Dependency tree · 2 levels

Sources

Karagila, Forcing & Symmetric Extensions, proof of Theorem 10.17