  A field k is called anti-Mordellic if every smooth curve over k with a
k-point has infinitely many k-points. We prove that for a function field over
an anti-Mordellic field, the subfield of constants is defined by a certain
universal first order formula. Under additional hypotheses regarding
2-cohomological dimension we prove that algebraic dependence of an n-tuple of
elements in such a function field can be described by a first order formula,
for each n. We also give a result that lets one distinguish various classes of
fields using first order sentences.
