  In partial answer to a question posed by Arnie Miller
(http://www.math.wisc.edu/~miller/res/problem.pdf) and X. Caicedo, we obtain
sufficient conditions for an L_{omega_1,omega} theory to have an independent
axiomatization. As a consequence we obtain two corollaries: The first, assuming
Vaught's Conjecture, every L_{omega_1,omega} theory in a countable language has
an independent axiomatization. The second, this time outright in ZFC, every
intersection of a family of Borel sets can be formed as the intersection of a
family of independent Borel sets.
