In partial answer to a question posed by Arnie Miller  and X. Caicedo  we obtain sufficient conditions for an ℒω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every ℒω1,ω 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.
Greg Hjorth. Ioannis A. Souldatos. "Independently axiomatizable ℒω1,ω theories." J. Symbolic Logic 74 (4) 1273 - 1286, December 2009. https://doi.org/10.2178/jsl/1254748691