DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Automorphisms acting on forcing names

Definition

A forcing automorphism $π : P \to P$ is a bijection preserving and reflecting the order, hence compatibility and incompatibility. Its action on names is defined by name-rank recursion:

$π \dot{x} = {⟨ π \dot{y}, π p ⟩ : ⟨ \dot{y}, p ⟩ \in \dot{x}} .$

Induction proves $(π σ) \dot{x} = π (σ \dot{x})$ , $π^{- 1} (π \dot{x}) = \dot{x}$ , and rank preservation. The same induction gives $π \overset{ˇ}{x} = \overset{ˇ}{x}$ for every ground set. Images of dense sets are dense, and if $G$ is generic then $π ‘ ‘ G$ is generic. No choice principle is required.

Depends on

Used by

Dependency tree · two levels

10 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, Definitions 10.1 and 10.5 (standard reference, not scraped)