  We present some Zermelo-Fraenkel consistency results regarding
bi-orderability of groups, as well as a construction of groups with Conradian
orders whose every action on metric spaces has bounded orbits. A classical
consequence of the ultrafilter lemma is that a group is bi-orderable if and
only if it is locally bi-orderable. We show that there exists a model of ZF in
which there is a group which is locally free (ergo locally bi-orderable) and
not bi-orderable, and the group can be given a total order. Such a group can
also exist in the presence of the principle of dependent choices. Comparable
consistency results are provided for torsion-free abelian groups.
