Halting problem, uncomputable sets: common mathematical proof?
The halting theorem, Cantor's theorem (the non-isomorphism of a set and its powerset), and Goedel's incompleteness theorem are all instances of the Lawvere fixed point theorem, which says that for any cartesian closed category, if there is an epimorphic map $e : A \to (A \Rightarrow B)$ then every $f : B \to B$ has a fixed point.


Lawvere, F. William. Diagonal Arguments and Cartesian Closed Categories. Lecture Notes in Mathematics, 92 (1969), 134-145.


For a nice introduction to these ideas, see this blog post of Andrej Bauer.