Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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Degree join is set union

Statement

For all A,B, the join of [A]T and [B]T is [A∪B]T.

Refutation

technique · direct

Given: Let 0′⊆N be an effective numerical coding of the halting language HALTTM.

[F1] The halting problem is undecidable (The halting problem is recognizable and undecidable), so the effectively coded set 0′ is not computable.

1.1givenconstruct

Take A=0′ and B=N∖0′. Since complementation is computable, [A]T=[B]T=[0′]T, and their degree join is [0′]T.

2.1F1step 1.1∎

But A∪B=N is computable, so [A∪B]T=[∅]T, which cannot equal [0′]T by [F1]. Hence ordinary union does not compute the degree join in general; tagged join is relevant.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources