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Early in the history of model theory, people like Tarski did extensive research on Boolean algebras. We know a lot about model theory of Boolean algebras from this era. It is not even hard to show quantifier elimination of the theory of atomless Boolean algebras. I believe that Boolean algebras are tame in this sense; they should be at least much nicer than models of PA, ZFC, etc.

Later on, in post-stability model theory, the theory of atomless Boolean algebras tended to be classified in the non-tame side of theories along with PA, ZFC, and the likes. The theory has an order and an independence in a quite trivial way, for instance. This seems to me a bit puzzling considering what I said in the first paragraph.

Does the more modern model theory has something to say regarding the tameness of Boolean algebras? Or the early results about them the only nice properties about them? (Perhaps other branches of logic like descriptive set theory might explain this issue; I am interested in an answer from such a perspective, too.)

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I'm not a model theorist, but my take is that different notions of tameness are looking for different things. For example, decidability is a lovely property of a theory from a computability-theoretic perspective, but it doesn't impact the number of models of a given cardinality the theory has. Even worse is true for quantifier elimination: we can always expand the language to get an equivalent theory with quantifier elimination, so q.e. doesn't tell us anything about the number of models.

If we're interested in classifying the models of a theory by "reasonable" invariants, or in something motivated by this, then things like decidability and quantifier elimination are likely to be irrelevant (at least in and of themselves - they may of course be very useful in analyzing a given theory or class of theories). So my two cents is that it's not that the theory of Boolean algebras turns out to be wild after all, but more that the notion of what "wildness" means changed significantly.

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