“S’il n’y a pas de solution, c’est qu’il n’y a pas de problème!”

Are some problems unsolvable?...

Gödel's incompleteness theorems state: (1) if you can list out the base assumptions of a system, then you cannot know all  logically true statements possible in the system or the system is inconsistent, i.e. contains a logical contradiction and (2) if you can use the system to prove the system is consistent - does not contain a logical contradiction - then the system must be inconsistent. There's a nice little rant here at the University of Michigan's website where a mathematician complaining about the misapplication of Gödel's incompleteness theorems ends up appearing to support intuitive reasoning over logical reasoning, i.e. that intuitive reasoning is more complete than logical reasoning.  I'm not sure if he's aware of this; I may pester him...