Coloring trees in reverse mathematics
Résumé
The tree theorem for pairs (TT 2 2), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2 <ω , there is a set of nodes isomorphic to 2 <ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT 2 2), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT 2 2 , by showing that this principle does not imply the arithmetic comprehension axiom (ACA 0) over the base system, recursive comprehension axiom (RCA 0), of second-order arithmetic. In addition , we give a new and self-contained proof of a recent result of Patey that TT 2 2 is strictly stronger than RT 2 2. Combined, these results establish TT 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA 0 and RT 2 2. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on ω.
Domaines
Logique [math.LO]
Origine : Fichiers produits par l'(les) auteur(s)
Loading...