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.

Nice names for subsets of a ground-model set

Definition

For a forcing order $P$ and ground-model set $A$ , a nice $P$ -name for a subset of $A$ is a name

$\dot{x} = {⟨ \overset{ˇ}{a}, p ⟩ : a \in A and p \in A_{a}},$

where each $A_{a} \subseteq P$ is an antichain. Empty $A_{a}$ are allowed, and if $A = \emptyset$ the displayed union is the empty name. Niceness alone does not assert that an arbitrary name is equivalent to such a name; that is the content of the next theorem.

Depends on

Used by

A nice name for one Cohen coordinate Example
Nice-name reduction and the ccc counting bound Theorem

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, nice-name reduction in the proof of Theorem 3.31 (standard reference, not scraped)