Independently axiomatizable ℒω1,ω theories

Journal of Symbolic Logic 74 (4):1273-1286 (2009)
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Abstract

In partial answer to a question posed by Arnie Miller [4] and X. Caicedo [2] 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

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