Interesting, God is the conceptual parallel of the statment "G" in Godel's incompleteness principle. This form in our minds is that which can't be defined. God is the concept which makes our logic incomplete, much like the statement "G" makes the system it is based upon incomplete.
God is what we cannot know, by its definition itself. It isn't what we don't know, it is the that which is outside the limitation of our state of mind and being.
It is impossible to deny the existence of God, because God is an implicit form in your own mind. However, by defining God as this, the existence of God does not effect anything in your life.