DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The classes Sigma_n^0, Pi_n^0, and Delta_n^0

Definition

For $n \geq 1$ , a set $A \subseteq N^{k}$ is $Σ_{n}^{0}$ when, for a primitive-recursive predicate $R$ , membership has a formula $\vec{x} \in A ⟺ \exists y_{1} \forall y_{2} \dots Q y_{n} R (\vec{x}, \vec{y}),$ where the quantifiers alternate and $Q$ is existential for odd $n$ . It is $Π_{n}^{0}$ when its complement is $Σ_{n}^{0}$ , equivalently when the display begins universally. Put $Δ_{n}^{0} : = Σ_{n}^{0} \cap Π_{n}^{0} .$

Bounded quantifiers in $R$ are allowed under the convention of Bounded arithmetic formulas.

Depends on

Bounded arithmetic formulas

Used by

Delta₁⁰ sets are exactly the decidable sets Corollary
A set lying in both Sigmaₙ⁰ and Piₙ⁰ Counterexample
Completeness at an arithmetical level Definition
False: Sigmaₙ⁰ and Piₙ⁰ are disjoint False statement
Prenex normalization preserves the arithmetical level Lemma
Post's theorem Theorem
Sigma₁⁰ sets are exactly the computably enumerable sets Theorem
The arithmetical hierarchy is strict Theorem
The totality set is Pi₂⁰-complete Theorem

Dependency tree · one level

1 result within one dependency step 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

Douglas Cenzer and Jeffrey Remmel, Effectively Closed Sets, Definition II.6.5 (standard reference, not scraped)