Notes from Hausdorff Institute Talk Oct 10, 2013

(a) isomorphism ∼=, elementary equivalence ≡, elementary equivalence ≡α for Lω1,ω sentences of rank < α. (b) Isomorphism of countable graphs, linear orders, countable Boolean algebras is ≤B complete for orbit equivalence relations of continuous S∞ actions (≤B is Borel reducibility, S∞ is the Polish group of permutations of ω). (c) for ∼= is partially answered in computable model theory, with notions such as relatively computably categorical, where presentations of X,Y together uniformly compute an isomorphism if there is one at all. For instance, a dense linear order is r.c.c.