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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.

03C75 Other infinitary logic
03C48 Abstract elementary classes and related topics
03E50 Continuum hypothesis and Martin’s axiom
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