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The nonabsoluteness of model existence in uncountable cardinals for $$L_{\omega_{1},\omega}$$. (English) Zbl 1285.03045
Summary: For sentences $$\varphi$$ of $$L_{\omega_{1},\omega}$$, we investigate the question of absoluteness of $$\varphi$$ having models in uncountable cardinalities. We first observe that having a model in $$\aleph_{1}$$ is an absolute property, but having a model in $$\aleph_{2}$$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis (GCH) context and provide sentences for any $$\alpha \in \omega_{1} \setminus\{0,1,\omega\}$$ for which the existence of a model in $$\aleph_{\alpha}$$ is nonabsolute (relative to large cardinal hypotheses). Finally, we present a complete sentence for which model existence in $$\aleph_{3}$$ is nonabsolute.

##### MSC:
 03C75 Other infinitary logic 03C48 Abstract elementary classes and related topics 03E50 Continuum hypothesis and Martin’s axiom
##### Keywords:
infinitary logic; absoluteness; abstract elementary classes
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##### References:
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