For a Hilbert space H, the (orthogonal) projections on H are those bounded operators given by orthogonal projection onto a closed subspace of H. Two such projections are equal modulo compact if they differ by a compact operator, that is, they have the same image in the Calkin algebra, the quotient of the bounded operators by the compact operators. Using Hjorth's theory of turbulence, we show that this equivalence relation is not classifiable by countable structures, and thus there are no (countable) algebraic complete invariants for the projections in the Calkin algebra. We also analyze the complexity of equivalence modulo finite rank operators, and show that this does not admit complete invariants given by the orbits of any Polish group action.