DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Numberings of partial computable functions

Definition

A numbering of the partial computable functions is a sequence $(φ_{e})_{e \in N}$ of partial computable functions $φ_{e} : N ⇀ N$ such that every partial computable unary function occurs as $φ_{e}$ for at least one $e \in N$ .

The number $e$ is then called an index or program number for the partial function $φ_{e}$ .

Remarks

A numbering need not be injective: different indices may compute the same partial function.
Later items treat a function of several arguments by applying a unary indexed function to a chosen code of the input tuple.

Depends on

A partial function is partial recursive if and only if it is Turing computable

Used by

Computable program transformers Definition
Productive and creative sets Definition
Universal and acceptable numberings Definition

Dependency tree · two levels

5 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

Robert I. Soare, Turing Computability: Theory and Applications (standard reference, not scraped)
Richard Zach, Incompleteness and Computability (standard reference, not scraped)