It turns out that these are all kind of the same, the Liar's Paradox, Russell's Paradox, the Halting Problem, etc. And indeed the they rely on a "diagonal map" i.e., a map x -> (x, x) in the construction.

This was a great read, if you want a mathematical take on it, and some generalization too. https://arxiv.org/abs/math/0305282v1