On the order dimension of locally countable partial orderings

We show that the order dimension of the partial order of all finite subsets of $\kappa$ under set inclusion is ${\log }_{}({\log }_{}(\kappa ))$ whenever $\kappa$ is an infinite cardinal. We also show that the order dimension of any locally countable partial ordering $(P,<)$ of size ${\kappa }^{+}$, for any $\kappa$ of uncountable cofinality, is at most $\kappa$. 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.