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


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 \(\aleph _1\).


Atomic models Pseudo-algebraic Non-structure 

Mathematics Subject Classification

Primary 03C50 Secondary 03C35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baldwin, J.T.: Categoricity. Number 51 in University Lecture Notes. American Mathematical Society, Providence, USA (2009)Google Scholar
  2. 2.
    Baldwin, J., Larson, P.: Iterated elementary embeddings and the model theory of infinitary logic. Ann. Pure Appl. Logic 167(3), 309–334 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baldwin, J., Laskowski, M.C., Shelah, S.: Constructing many uncountable atomic models in \(\aleph _1\). J. Symb. Log. 81(3), 1142–1162 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baldwin, J., Laskowski, M.C.: Henkin constructions of models of size continuum. Bull. Symb. Log. (to appear)Google Scholar
  5. 5.
    Baldwin, J., Laskowski, M.C., Shelah, S.: An active chain implies many atomic models in \(\aleph _1\) (preprint)Google Scholar
  6. 6.
    Keisler, H.J.: Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers. Studies in Logic and the Foundations of Mathematics, vol. 62. North-Holland Publishing Co., Amsterdam (1971)zbMATHGoogle Scholar
  7. 7.
    Keisler, H.J.: Logic with the quantifier there exist uncountably many. Ann. Math. Log. 1, 1–93 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shelah, S.: Classification Theory and the Number of Nonisomorphic Models. North-Holland, Amsterdam (1978)Google Scholar
  9. 9.
    Shelah, S.: How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null. Isr. J. Math. 88, 159–174 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ulrich, D., Rast, R., Laskowski, M.C.: Borel complexity and potential canonical Scott sentences. Fundam. Math. Fundam. Math. 239(2), 101–147 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations