TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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Kleene's second recursion theorem

Statement

Fix an acceptable numbering $(φ_{e})_{e \in N}$ . For every computable program transformer $F : N \to N$ , there exists an index $e$ such that $φ_{e} = φ_{F (e)}$ as partial functions.

Facts & Assumptions

Given: A computable program transformer $F$ .

[L1]

A computable program transformer is a total computable map on indices, by Computable program transformers.

[L2]

There is a total computable map $d$ with $φ_{d (u)} (x) ≃ φ_{u} (⌜ u, x ⌝_{s e q}),$ by The diagonal self-reference construction from s-m-n.

Proof

technique · direct

1.1

Let $d$ be the diagonal map from [L2]. Because $F$ is total computable by [L1], the binary partial function $Ψ (z, x) : = φ_{F (d (z))} (x)$ is partial computable. Since we are working inside a numbering of all partial computable functions, choose an index $q$ with $φ_{q} (⌜ z, x ⌝_{s e q}) ≃ Ψ (z, x)$ for all $z, x$ .

L1L2givenchoose

2.1

Set $e : = d (q)$ . Then for every input $x$ , $φ_{e} (x) ≃ φ_{q} (⌜ q, x ⌝_{s e q}) ≃ Ψ (q, x) = φ_{F (d (q))} (x) = φ_{F (e)} (x),$ where the first equivalence is [L2] and the last equality is the definition of $e$ . Hence $φ_{e} = φ_{F (e)}$ .

L2step 1.1algebra

3.1

Therefore every computable program transformer has a fixed point.

step 2.1∎

Depends on

Used by

FALSE: a partial computable function has a unique program index False statement
FALSE: the recursion theorem requires source-code access False statement
Acceptable numberings are computably intertranslatable Proposition
The recursion theorem with parameters 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

Lawrence S. Moss, Invitation to Computability and Recursion, The Recursion Theorem (standard reference, not scraped)
Robert I. Soare, Turing Computability: Theory and Applications (standard reference, not scraped)