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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 229 / 2015

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

#### Streszczenie

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.

#### Autorzy

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