Isomorphism of countable structures and infinitary logic

**Abstract:**

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 *L*_{ω1ω} 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 *L*_{ω1ω} (or even
an "analytic" over *L*_{ω1ω}
sentence) is either ≤ ℵ_{0}, or equal to
ℵ_{1}, or equal to 2^{ℵ0}.