The story reads great :D But first and foremost, I like how ``it cannot be proven that...'' is a special kind of negation, much different than the ordinary neg

The story reads great :D But first and foremost, I like how ``it cannot be proven that...'' is a special kind of negation, much different than the ordinary negation in boolean logic. Is there any formalized logic system with a negation with properties like ``it cannot be proven that...''?