False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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False: a limit-computable function has a known stabilization stage

Statement

Every limit-computable function has a computable modulus of stabilization.

Facts & Assumptions

Given: the standard stage approximation to the halting set.

Refutation

technique · contradiction

1.1

Assume a computable modulus is supplied for this approximation. Let $h (e, s) = 1$ if program $e$ has halted by stage $s$ , and $0$ otherwise. This is computable and converges pointwise to $χ_{0^{'}}$ .

givenassume-contraconstruct

2.1

If a computable modulus $m (e)$ were available, simulate $e$ for $m (e)$ steps and return $h (e, m (e))$ . This would decide $0^{'}$ , contradicting undecidability of the halting problem.

step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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, §4.7 (standard reference, not scraped)