CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-02

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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.

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The busy beaver function is not computable

Statement

The busy beaver function $B B : N \to N$ is not computable.

Facts & Assumptions

Given: The busy beaver function $B B$ .

[L1]

Every total computable function is eventually dominated by $B B$ , by The busy beaver function eventually dominates every total computable function.

[L2]

"Computable" means total computable in the sense of Computable and partial computable functions.

[L3]

The function named $B B$ is the busy beaver function of The step-count busy beaver function relative to the chosen machine coding.

Proof

technique · direct

1.1

Suppose for contradiction that $B B$ were computable in the sense of [L2]. Then the function $g (n) : = B B (n) + 1$ would also be total computable.

assume-contraL2L3algebra

2.1

Applying [L1] to the computable function $g$ yields an $N$ such that $B B (n) \geq g (n) = B B (n) + 1$ for every $n \geq N$ . That is impossible.

L1step 1.1contradiction

3.1

Therefore $B B$ is not computable.

step 2.1discharge-contradiction: a computable busy beaver would dominate itself plus one∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Adam Yedidia and Scott Aaronson, A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory (standard reference, not scraped)