DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Completeness at an arithmetical level

Definition

Fix the canonical binary-numeral encoding $ν : N \to {0, 1}^{*}$ , and write $ν (A) = {ν (a) : a \in A}$ . For $A, B \subseteq N$ , write $A \leq_{m} B$ when $ν (A) \leq_{m} ν (B)$ by a total computable string function in the sense of Computable many-one reductions between languages.

For $n \geq 1$ , $B \subseteq N$ is $Σ_{n}^{0}$ -complete when $B \in Σ_{n}^{0}$ and every $A \subseteq N$ with $A \in Σ_{n}^{0}$ satisfies $A \leq_{m} B$ . Define $Π_{n}^{0}$ -completeness identically with $Π_{n}^{0}$ in place of $Σ_{n}^{0}$ .

Depends on

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

Used by

The halting set is Sigma₁⁰-complete Theorem
The totality set is Pi₂⁰-complete Theorem

Dependency tree · two levels

2 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

Ludovic Patey, Computability Theory, Definition 5.2 (standard reference, not scraped)