Impossibility
Careful use of language dictates that we distinguish between several senses in which something can be said to be impossible. Some things are logically impossible: you can't have a square circle or an object that is both perfectly black and perfectly not-black. Also, in our reductionist universe operating according to universal physical laws, some things are physically impossible based on our model of how things work, even they are not obviously contradictory or contrary to reason: for example, the laws of thermodynamics give us a strong guarantee that there can never be a perpetual motion machine.
It can be tempting to label as impossible very difficult problems which you have no idea how to solve. But the apparent lack of a solution is not a strong guarantee that no solution can exist in the way that the laws of thermodynamics, or Godel's incompleteness results, give us proofs that something can not be accomplished. A blank map does not correspond to a blank territory; in the absence of a proof that a problem is insolvable, you can't be confident that you're not just overlooking something that a greater intelligence would spot in an instant.