Unsolvable

The unsolvability of polynomial equations of degree greater than 5 by radicals as well as Gödel's incompleteness theorem are examples of results that show the definitive impossibility of solving a problem, no matter which way we try it. 

Which novel thoughts come into such an unsolvability result, and how are they perceived, both within and outside the mathematical world? Discuss this with respect to a suitable example, but not one of the above.