  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.
