Preprints, Working Papers, ... Year : 2025

Post's Problem in Constructive Mathematics

Abstract

We study a solution to Post's problem, i.e. the existence of a semi-decidable but undecidable Turing degree strictly below the halting problem, from the perspective of constructive mathematics. We use the finite injury priority method due to Friedberg and Muchnik to carry out the construction of a low simple set due to Soare.

Our proof reveals that as only non-constructive logical principle it suffices to assume (¬¬Σ1)-LEM, stating that for any Σ1 set A, ¬¬∀x. x ∈ A ∨ x ̸ ∈ A. In usual settings of constructive reverse mathematics, this principle is classified as strictly weaker than LEM, LPO, or even ¬¬LPO.

Our proof is explained in (analytic) textbook computability theory based on a model of computation, but additionally formally backed by a machine-checked development in synthetic computability using the Rocq proof assistant.

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Dates and versions

hal-04923365 , version 1 (31-01-2025)

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  • HAL Id : hal-04923365 , version 1

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Haoyi Zeng, Yannick Forster, Dominik Kirst, Takako Nemoto. Post's Problem in Constructive Mathematics. 2025. ⟨hal-04923365⟩
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