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

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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 \(\aleph _1\).

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, USA

    Michael C. Laskowski

  2. Hebrew University, Jerusalem, Israel

    Saharon Shelah

  3. Rutgers University, New Brunswick, USA

    Saharon Shelah

Authors
  1. Michael C. Laskowski
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  2. Saharon Shelah
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Corresponding author

Correspondence to Michael C. Laskowski.

Additional information

M. C. Laskowski: Partially supported by NSF Grant DMS-1308546.

S. Shelah: Partially supported by European Research Council Grant 338821 and NSF Grant DMS-1362974. Publication no. 1099.

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Laskowski, M.C., Shelah, S. A strong failure of \(\aleph _0\)-stability for atomic classes. Arch. Math. Logic 58, 99–118 (2019). https://doi.org/10.1007/s00153-018-0623-6

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  • DOI: https://doi.org/10.1007/s00153-018-0623-6

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