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## Fundamenta Mathematicae

2007 | 195 | 3 | 269-289
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### The effective Borel hierarchy

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Let K be a subclass of Mod(𝓛) which is closed under isomorphism. Vaught showed that K is $Σ_α$ (respectively, $Π_α$) in the Borel hierarchy iff K is axiomatized by an infinitary $Σ_α$ (respectively, $Π_α$) sentence. We prove a generalization of Vaught's theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective $Σ_α$ or effective $Π_α$ Borel set with a computable infinitary sentence of the same complexity. This result yields an alternative proof of Vaught's theorem via forcing. We also get a version of the pull-back theorem from Knight et al. which says if Φ is a Turing computable embedding of K ⊆ Mod(𝓛) into K' ⊆ Mod(ℒ'), then for any computable infinitary sentence φ in the language 𝓛, we can find a computable infinitary sentence φ* in 𝓛' such that for all 𝓐 ∈ K, 𝓐 ⊨ φ* iff Φ(𝓐 ) ⊨ φ, where φ* has the same complexity as φ.
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Tom
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269-289
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2007
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• Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.
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