1,439 research outputs found
Reverse mathematics and infinite traceable graphs
This paper falls within the general program of investigating the proof
theoretic strength (in terms of reverse mathematics) of combinatorial
principals which follow from versions of Ramsey's theorem. We examine two
statements in graph theory and one statement in lattice theory proved by
Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our
main results are that the statements concerning graph theory are equivalent to
Ramsey's theorem for 4-tuples over \RCA while the statement concerning
lattices is provable in \RCA.
Revised 12/2010. To appear in Archive for Mathematical Logi
Î 10 classes and orderable groups
AbstractIt is known that the spaces of orders on orderable computable fields can represent all Î 10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Î 10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes
Lowness notions, measure and domination
We show that positive measure domination implies uniform almost everywhere
domination and that this proof translates into a proof in the subsystem
WWKL (but not in RCA) of the equivalence of various Lebesgue measure
regularity statements introduced by Dobrinen and Simpson. This work also allows
us to prove that low for weak -randomness is the same as low for
Martin-L\"of randomness (a result independently obtained by Nies). Using the
same technique, we show that implies , generalizing the
fact that low for Martin-L\"of randomness implies low for
The complexity of central series in nilpotent computable groups
AbstractThe terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders
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