Non-Wadgean invariant descriptive set theory
Consider a continuous action a by a Polish group G on a Polish
space X. An important special case of this is the logic action for the
language L: the group of permutations of N acts on codes for countable
L-structures, and the orbit equivalence relation is isomorphism. The
subject of "invariant descriptive set theory", which originates with
Vaught in 1974, involves generalizing concepts and theorems from ordinary
descriptive set theory (the case of G being the trivial group) to this
situation. It is the thesis of this talk that we have been doing it wrong
for 24 years. The proper analog of analytic sets is not invariant analytic
sets, but rather, what I call "satisfactory" sets. In the special case of
logic actions, a subset S of X is satisfactory iff S is analytic and for
some countable fragment F of infinitary logic, S is closed under
F-embeddings. For a-invariant sets, the pointclass of satisfactory sets is
properly between the classes of Borel and analytic sets, i.e., it is
non-Wadgean. Vaught's Conjecture for satisfactory sets is open.