Degrees bounding principles and universal instances in reverse mathematics

Ludovic Patey (1, 2)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 PI.R2 - Design, study and implementation of languages for proofs and programs

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) do not admit a bound of low 2 degree. Therefore no principle between Ramsey's theorem for pairs (RT 2 2) and SADS or STS(2) admit a universal instance. We construct a low 2 degree bounding the Erd˝ os Moser theorem (EM), thereby showing that the previous argument does not hold for EM. Finally, we prove that the only ∆ 0 2 degree bounding a stable version of the rainbow Ramsey theorem for pairs (SRRT 2 2) is 0. Hence no principle between the stable Ramsey theorem for pairs (SRT 2 2) and SRRT 2 2 admit a universal instance. In particular the stable version of the Erd˝ os-Moser theorem does not admit one. It remains unknown whether EM admits a universal instance.

Domaines: Logique [math.LO]

Ludovic Patey. Degrees bounding principles and universal instances in reverse mathematics. Annals of Pure and Applied Logic, 2015, 166 (11), pp.1165 - 1185. ⟨10.1016/j.apal.2015.07.003⟩. ⟨hal-01888599⟩