Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 58, Number 2 (2017), 159-177.
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.
Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 159-177.
Received: 2 February 2012
Accepted: 1 September 2014
First available in Project Euclid: 28 January 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48]
Secondary: 03C55: Set-theoretic model theory
Shelah, Saharon. Universal Structures. Notre Dame J. Formal Logic 58 (2017), no. 2, 159--177. doi:10.1215/00294527-3800985. https://projecteuclid.org/euclid.ndjfl/1485572517