Borel completeness of some $\aleph _{0}$-stable theories
Volume 229 / 2015
Fundamenta Mathematicae 229 (2015), 1-46
MSC: Primary 03C45; Secondary 03E15.
DOI: 10.4064/fm229-1-1
Abstract
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.
