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Borel completeness of some $\aleph _{0}$-stable theories

Volume 229 / 2015

Michael C. Laskowski, Saharon Shelah Fundamenta Mathematicae 229 (2015), 1-46 MSC: Primary 03C45; Secondary 03E15. DOI: 10.4064/fm229-1-1


We study $\aleph _0$-stable theories, and prove that if $T$ either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of $\lambda $-Borel completeness and prove that such theories are $\lambda $-Borel complete. Using this, we conclude that an $\aleph _0$-stable theory satisfies $I_{\infty ,\aleph _0}(T,\lambda )=2^\lambda $ for all cardinals $\lambda $ if and only if $T$ either has eni-DOP or is eni-deep.


  • Michael C. LaskowskiDepartment of Mathematics
    University of Maryland
    College Park, MD 20742, U.S.A.
  • Saharon ShelahDepartment of Mathematics
    The Hebrew University of Jerusalem
    Einstein Institute of Mathematics
    Edmond J. Safra Campus, Givat Ram
    Jerusalem, 91904, Israel
    Department of Mathematics
    Hill Center, Busch Campus
    Rutgers, the State University of New Jersey
    110 Frelinghuysen Road
    Piscataway, NJ 08854-9019, U.S.A.

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