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

Michael C. Laskowski ¹   ; Saharon Shelah ²

¹ Department of Mathematics University of Maryland College Park, MD 20742, U.S.A.
² Department 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 110 Frelinghuysen Road Piscataway, NJ 08854-9019, U.S.A.

Fundamenta Mathematicae, Tome 229 (2015) no. 1, pp. 1-46

Résumé

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.

Détail

DOI : 10.4064/fm229-1-1

Keywords: study aleph stable theories prove either has eni dop eni deep its class countable models borel complete introduce notion lambda borel completeness prove theories lambda borel complete using conclude aleph stable theory satisfies infty aleph lambda lambda cardinals lambda only either has eni dop eni deep