I've been wondering how an argument that a solution to a particular problem doesn't exist is put together.
For instance "Pour-El and Richards found an ordinary differential equation $\phi'(t)=F(t,\phi(t))$ with $F$ computable and no computable solution."
Another puzzling example is the non-computability of Chaitin's constant.
I'm not talking about trivial examples, e.g. the DE $f(x)=f(x)+1$.
Could you give examples of outlines of such proofs? The only strategy I can think of is showing that the assumption that a solution exists leads to a contradiction, but surely there are more sophisticated ways than this?
EDIT: In case it wasn't clear (it probably wasn't), I also would examples of proof strategies involved in proving the non-existence of solutions to, for instance, DEs. The reason my examples focuses on computability is because I couldn't find any examples of the non-existence.