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.

Cohen, collapse, and Lévy-collapse forcing orders

Definition

For infinite $κ$ and nonzero $λ$ ,

$Add (κ, λ) = Fn (λ \times κ, 2, < κ),$

the partial functions of domain size $< κ$ , ordered by reverse inclusion. For infinite $κ \leq λ$ put

$Col (κ, λ) = Fn (κ, λ, < κ) .$

For an ordinal $θ$ , the finite-condition Lévy order $Lv (θ)$ consists of finite functions $p$ with $dom p \subseteq θ \times ω$ and $p (α, n) < α$ whenever $(α, n) \in dom p$ . In all three cases the empty function is largest and compatible conditions have union as a common extension. These are ground-model sets when used as forcing orders. The second order adds one $κ$ -indexed surjection onto $λ$ ; the third simultaneously addresses every nonzero $α < θ$ .

Depends on

Dependency tree · 2 levels

Sources

Karagila, Forcing & Symmetric Extensions, Chapters 3–4