4 research outputs found
Degrees bounding principles and universal instances in reverse mathematics
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 RT22 and SADS or STS(2) admit a universal
instance. We construct a low_2 degree bounding the Erd\H{o}s-Moser theorem
(EM), thereby showing that previous argument does not hold for EM. Finally, we
prove that the only Delta^0_2 degree bounding a stable version of the rainbow
Ramsey theorem for pairs (SRRT22) is 0'. Hence no principle between the stable
Ramsey theorem for pairs SRT22 and SRRT22 admit a universal instance. In
particular the stable version of the Erd\H{o}s-Moser theorem does not admit
one. It remains unknown whether EM admits a universal instance.Comment: 23 page
Dominating the Erdos-Moser theorem in reverse mathematics
The Erdos-Moser theorem (EM) states that every infinite tournament has an
infinite transitive subtournament. This principle plays an important role in
the understanding of the computational strength of Ramsey's theorem for pairs
(RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the
ascending descending sequence principle (ADS). In this paper, we study the
computational weakness of EM and construct a standard model (omega-model) of
simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which
is not a model of the atomic model theorem. This separation answers a question
of Hirschfeldt, Shore and Slaman, and shows that the weakness of the
Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman,
Solomon and Towsner.Comment: 36 page
Models of Arithmetic and Subuniform Bounds for the Arithmetic Sets
It has been known for more than thirty years that the degree of a nonstandard model of true arithmetic is a subuniform upper bound for the arithmetic sets (suub). Here a notion of generic enumeration is presented with the property that the degree of such an enumeration is an suub but not the degree of a non-standard model of true arithmetic. This anwers a question posed in the literature
Actas de las XXXIV Jornadas de Automática
Postprint (published version