Independently axiomatizable ℒω1,ω theories

Abstract

In partial answer to a question posed by Arnie Miller [4] and X. Caicedo [2] we obtain sufficient conditions for an ℒ_ω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every ℒ_ω1,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.