We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or . We use versions of being reduced—replacing by a subring (defined by a sequence )—and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with (variants of) the oak property (from a work of Džamonja and the author), a property of complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier-free formulas) and deals more with the existence of universals.
References
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[1] Džamonja, M., “Club guessing and the universal models,” Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 283–300. 1105.03037 10.1305/ndjfl/1125409327 euclid.ndjfl/1125409327
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[20] Shelah, S., Classification Theory for Abstract Elementary Classes, vol. 18 of Studies in Logic: Mathematical Logic and Foundations, College Publications, London, 2009. 1225.03036[20] Shelah, S., Classification Theory for Abstract Elementary Classes, vol. 18 of Studies in Logic: Mathematical Logic and Foundations, College Publications, London, 2009. 1225.03036
[22] Shelah, S., “Abstract elementary classes near $\aleph_{1}$,” preprint, arXiv:0705.4137v1 [math.LO]. 0705.4137v1[22] Shelah, S., “Abstract elementary classes near $\aleph_{1}$,” preprint, arXiv:0705.4137v1 [math.LO]. 0705.4137v1
[23] Shelah, S., “Black boxes,” preprint, arXiv:0812.0656v2 [math.LO]. 0812.0656v2 MR3184377 1314.03042[23] Shelah, S., “Black boxes,” preprint, arXiv:0812.0656v2 [math.LO]. 0812.0656v2 MR3184377 1314.03042
[24] Shelah, S., “General non-structure theory,” preprint, arXiv:1011.3576v2 [math.LO]. 1011.3576v2[24] Shelah, S., “General non-structure theory,” preprint, arXiv:1011.3576v2 [math.LO]. 1011.3576v2
[25] Shelah, S., “Model theory for a compact cardinal,” to appear in Israel Journal of Mathematics, preprint, arXiv:1303.5247v3 [math.LO]. 1303.5247v3[25] Shelah, S., “Model theory for a compact cardinal,” to appear in Israel Journal of Mathematics, preprint, arXiv:1303.5247v3 [math.LO]. 1303.5247v3
