  We combine computable structure theory and algorithmic learning theory to
study learning of families of algebraic structures. Our main result is a
model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$,
consisting of the structures whose isomorphism types can be learned in the
limit. We show that a family of structures $\mathfrak{K}$ is
$\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from
$\mathfrak{K}$ can be distinguished in terms of their
$\Sigma^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar
cases and we show the following: there is an infinite learnable family of
distributive lattices; no pair of Boolean algebras is learnable; no infinite
family of linear orders is learnable.
