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A LOPEZ-ESCOBAR THEOREM FOR CONTINUOUS DOMAINS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 15 March 2024, pp. 1-18
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We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let
$Mod(\tau )$ be the set of countable structures with universe 
$\omega $ in vocabulary 
$\tau $ topologized by the Scott topology. We show that an invariant set 
$X\subseteq Mod(\tau )$ is 
$\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a 
$\Pi ^p_\alpha $-formula, a positive 
$\Pi ^0_\alpha $ formula in the infinitary logic 
$L_{\omega _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let 
$\mathcal {K}$ be positively computably embeddable in 
$\mathcal {K}'$ by 
$\Phi $, then for every 
$\Pi ^p_\alpha $ formula 
$\xi $ in the vocabulary of 
$\mathcal {K}'$ there is a 
$\Pi ^p_\alpha $ formula 
$\xi ^{*}$ in the vocabulary of 
$\mathcal {K}$ such that for all 
$\mathcal {A}\in \mathcal {K}$, 
$\mathcal {A}\models \xi ^{*}$ if and only if 
$\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.
