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Vaught's Conjecture has been a fundamental motivating question in Model Theory for decades now; the solutions to various special cases all seem to support the conjecture, but also indicate that a full solution will be extremely difficult.

A complete finitary first order theory in a countable language must have no more than continuum many countable models up to isomorphism (a simple counting argument in set theory); Robert Vaught proved in the late 1950's that such a theory cannot have precisely two countable models. Any other finite number of models is possible. There are easy examples of countable theories with countably many and with continuum many countable models.

Now our choice of Set Theory enters in: it may be that the continuum is much larger than ℵ₀. Morley proved that the number of countable models of a complete countable theory is either the continuum, or less than or equal to ℵ₁. Vaught's Conjecture is that the number of countable models of a complete countable theory is either the continuum, or less than or equal to ℵ₀.

Apparently the only viable approach to solving this problem is to show that if such a theory does not have many models (that is, continuum many models) then at least the countable models of the theory must satisfy some kind of structure theorem, which enables us to count the isomorphism classes and prove that there are only countably many. This process has been carried out for many special classes of theories, where some kind of structure theory is already known, including theories of modules over certain special kinds of rings.

In a recent paper (Vaught's conjecture for superstable theories of finite rank in Annals of Pure and Applied Logic), Steven Buechler has provided the most general solution so far to part of Vaught's conjecture. The proof is of incredible technical complexity, but in the end one of the important features is to be able to understand the structure of certain kinds of modules.

In this talk I will give a very informal overview of this subject area and some of the ideas involved. In a later talk I hope to discuss some of the specific details of interest about modules.