Responding to the topic at hand, paradoxes are not meant to be evaluated a second time. In other words, a paradox can make excellent sense until you go back and read it again, having already read the paradox once.

Example: This sentence is false.

The above sentence is false. If you can't read it again, you can't apply the knowledge you already have to try to decipher it further. It's false. Live with it :P

This theorem cannot be proven.

Correct. You can't prove that something can't be proven. So, the above theorem is unprovable. Since that's explicitly stated in the text of the theorem, there is no proof.