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Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 99–118 | Cite as

A strong failure of \(\aleph _0\)-stability for atomic classes

  • Michael C. LaskowskiEmail author
  • Saharon Shelah
Article
  • 19 Downloads

Abstract

We study classes of atomic models \(\mathbf{At}_T\) of a countable, complete first-order theory T. We prove that if \(\mathbf{At}_T\) is not \(\mathrm{pcl}\)-small, i.e., there is an atomic model N that realizes uncountably many types over \(\mathrm{pcl}_N(\bar{a})\) for some finite \(\bar{a}\) from N, then there are \(2^{\aleph _1}\) non-isomorphic atomic models of T, each of size \(\aleph _1\).

Keywords

Atomic models Pseudo-algebraic Non-structure 

Mathematics Subject Classification

Primary 03C50 Secondary 03C35 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Hebrew UniversityJerusalemIsrael
  3. 3.Rutgers UniversityNew BrunswickUSA

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