Let $S$ denote de cantor set on $[0,1]$ then $S$ is non enumerable.
Sketch: Assume by contradiction that $S$ is enumerable then by additive theory of sets the sum $A+B$ = {$a+b: a\in A, b\in B$} is enumerable if both $A,B$ are enumerable,but then $S+S=[0,2]$is non-enumerable,contradiction.
Is that a reasonable alternative for the Cantor's diagonal proof?