The descriptive set theory of C^*-algebra invariants

We establish the Borel computability of various C$^*$-algebra invariants, including the Elliott invariant and the Cuntz semigroup. As applications we deduce that AF algebras are classifiable by countable structures, and that a conjecture of Winter and the second author for nuclear separable simple C*-algebras cannot be disproved by appealing to known standard Borel structures on these algebras.