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We study aleph_0-stable theories, and prove that if T either has eni-DOP oris eni-deep, then its class of countable models is Borel complete. We introducethe notion of lambda-Borel completeness and prove that such theories arelambda-Borel complete. Using this, we conclude that an aleph_0-stable theoryhas 2^lambda pairwise non-L(infinity,aleph_0) equivalent models of size lambdafor all infinite cardinals lambda if and only if T either has eni-DOP or iseni-deep.

Borel completeness of some aleph_0 stable theories