>>11313202They are known as diagonalization arguments. The intuition in this case is that if you have -many subsets of , view these each as binary strings of length (1 meaning in the set, 0 not in the set), and stack these up to form an grid. Then if you take the string formed along the diagonal, and flip each of its bits, this new string is different from all the strings you started with. (And therefore represents a subset different from the subsrts you started with.) All the theorems you listed have a step that resembles this.
There is also a general abstract result in category theory called Lawvere's theorem which sort of ties them together but it's disappointing in some ways and, being category theory, actually kinda gay
