No such experiment CAN exist. Re-stating Gödel's First Incompleteness Theorem in this context would say that in order to absolutely prove or disprove the existence of God, you would have to do so from the context of a being/system greater than God. In other words, to prove or disprove the existence of God, you have to believe that an even higher authority exists.
This is why faith is critical to the process.
Sorry, but I should have stated in my initial response that the response was rhetorical. Just as you can’t define a word using that same word, you also can’t define GOD unless you believe in an authority greater than GOD.