Impossibility theorems I've been wondering how you go about proving an impossibility e.g. when I looked up Abel's impossibility theorem it says nothing about the proof and only restates the theorem when I'd like to know how it's done. Is the way to prove an impossibility that you assume the opposite - that it's possible - and derive a contradiction from the possibility and hence you have proven an impossibility? Thank you

Proving "not $P$" can be done by showing two things: "if not $Q$, then not $P$" and "not $Q$". But this is equivalent to proving the contrapositive, "if $P$ then $Q$", along with "not $Q$". The latter is what we call "proof by contradiction" (assuming the opposite).

The Wikipedia article gives a proof of Abel-Ruffini which boils down to "If the Galois group of $S_n$ is not solvable, then general $n^{th}$ degree polynomials are not solvable by radicals", together with "$S_5$ is not a solvable group". This particular proof is not technically "proof by contradiction" (but is equivalent to such a proof).