DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedverified 2026-09-08 (gpt-6-astra)

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

For $A, B \subseteq N$ , write $A \leq_{m} B$ when there is a total computable function $f : N \to N$ such that $\forall a \in N, a \in A ⟺ f (a) \in B .$ Here total computability means Computable and partial computable functions, using canonical binary numerals to represent natural numbers. In particular, the machine must halt with a valid numeral output on every numeral input. This condition concerns functions on natural numbers; an invalid numeral is not an extra possible output value.

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

Used by

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

Dependency tree · two levels

4 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)