Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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The halting oracle is not computably dominated

Statement

The ordinary halting oracle 0′=∅′ is not computably dominated.

Facts & Assumptions

Given: the halting oracle 0′ and an arbitrary total computable candidate dominator g.

Proof

technique · direct
1.1

Using 0′, define h(e) to be 0 if program e does not halt on e, and otherwise its least halting stage on e. This is total 0′-computable.

givenconstruct
1.2

The recursion theorem supplies an index e for a program which on its own input computes g(e) and then performs g(e)+1 dummy steps before halting. Hence h(e)≥g(e)+1.

givenconstruct
2.1

Thus arbitrary computable g fails to dominate h. No computable function dominates every total 0′-computable function, so 0′ is not computably dominated.

step 1.1step 1.2∎

Depends on

Used by

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