This is intended as an introductory talk to a large but mainly old (40 years old or so) subject that does not seem to be very active at the moment (I can always be wrong with such statements: I am not working in the area now (except in a marginal way), and I am not following the literature; my information on present-day activity comes by hearsay). However, the subject mentioned in the title is closely related to descriptive set-theory, and in particular, to (complexity) theory of equivalence relations. In the introductory talk, I will explain the basics of infinitary logic, and try to go towards proofs of two theorems, 1. and 2. as follows:
1. The Ryll-Nardzewski/Dana Scott theorem: the isomorphism class of a single countable structure is Lω1ω definable (the Scott version); or equivalently, the minimal non-empty isomorphism-invariant sets of countable structures with underlying set the integers are Borel (the Ryll-Nardzewski version).
2. Morley's theorem: the number of isomorphism classes of countable models of a sentence of Lω1ω (or even an "analytic" over Lω1ω sentence) is either ≤ ℵ0, or equal to ℵ1, or equal to 2ℵ0.