On the order dimension of locally countable partial orderings

Abstract

We show that the order dimension of any locally countable partial ordering ( P , > ) (P, >) of size κ + \kappa ^+ , for any κ \kappa of uncountable cofinality, is at most κ \kappa . In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.