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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
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[1] Baldwin, J. T., “Amalgamation, absoluteness and categoricity,” pp. 22-50 in Proceedings of the 11th Asian Logic Conference , World Science Publishing, Hackensack, N.J., 2012. · Zbl 1278.03073
[2] Barwise, J., “The role of the omitting types theorem in infinitary logic,” Archiv für Mathematische Logik und Grundlagenforschung , vol. 21 (1981), pp. 55-68. · Zbl 0467.03034
[3] Chang, C. C., and H. J. Keisler, Model Theory , 3rd edition, vol. 73 of Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 1990. · Zbl 0697.03022
[4] Cummings, J., M. Foreman, and M. Magidor, “Squares, scales and stationary reflection,” Journal of Mathematical Logic , vol. 1 (2001), pp. 35-98. · Zbl 0988.03075
[5] Gao, S., “On automorphism groups of countable structures,” Journal of Symbolic Logic , vol. 63 (1998), pp. 891-96. · Zbl 0922.03045
[6] Gregory, J., “Elementary extensions and uncountable models for infinitary finite quantifier language fragments,” Notices of the American Mathematical Society , vol. 17 (1970), pp. 967-68.
[7] Grossberg, R., and S. Shelah, “On the number of nonisomorphic models of an infinitary theory which has the infinitary order property, I,” Journal of Symbolic Logic , vol. 51 (1986), pp. 302-22. · Zbl 0631.03022
[8] Hjorth, G., “Knight’s model, its automorphism group, and characterizing the uncountable cardinals,” Journal of Mathematical Logic , vol. 2 (2002), pp. 113-44. · Zbl 1010.03036
[9] Jech, T., Set Theory , 3rd millennium edition, revised and expanded, Springer Monographs in Mathematics , Springer, Berlin, 2003.
[10] Jensen, R. B., “The fine structure of the constructible hierarchy,” with a section by Jack Silver, Annals of Mathematical Logic , vol. 4 (1972), pp. 229-308. · Zbl 0257.02035
[11] Keisler, H. J., Model Theory for Infinitary Logic: Logic with Countable Conjunctions and Finite Quantifiers , vol. 62of Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 1971. · Zbl 0222.02064
[12] Malitz, J., “The Hanf number for complete \(L_{\omega_{1},\omega}\)-sentences,” pp. 166-81 in The Syntax and Semantics of Infinitary Languages , edited by J. Barwise, vol. 72 of Lecture Notes in Mathematics , Springer, Berlin, 1968.
[13] Specker, E., “Sur un problème de Sikorski,” Colloquium Mathematicum , vol. 2 (1949), pp. 9-12. · Zbl 0040.16703
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