Borel completeness of some aleph_0 stable theories

TL;DR

This paper proves that certain aleph_0-stable theories with eni-DOP or eni-deep properties have highly complex classification problems, characterized by Borel completeness, leading to many non-equivalent models of any infinite size.

Contribution

It introduces the notion of lambda-Borel completeness and establishes a criterion linking eni-DOP or eni-deep properties to the complexity of models in aleph_0-stable theories.

Findings

Theories with eni-DOP or eni-deep are Borel complete.

Such theories have 2^lambda non-L-infinity,aleph_0 equivalent models of size lambda.

Lambda-Borel completeness is a key concept introduced in this work.

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 has 2^lambda pairwise non-L(infinity,aleph_0) equivalent models of size lambda for all infinite cardinals lambda if and only if T either has eni-DOP or is eni-deep.