Here are real example of Incompleteness Theorem used to reveal the fundamental inconsistency in the God Hypothesis.

Can God construct a building he can't tear down?

If God did indeed create the Universe, then he is obviously no longer needed. Why should God need to continuously interfere with the Universe, manage it and tweak it? Couldn't he get it right the first time? And if he couldn't, then he's not much of a God, is he?

If you can't use logic to prove or disprove God's existence, then that statement itself cannot be logically proven or disproven. So there really is no point in talking about it, is there?

Just because you use fancy language and quote authoritative thinkers doesn't mean you can draw any "scientific" conclusions about God.

We do not know whether or not there ever was a God.

If there was a God, we do not know whether or not he still exists.

If he exists, we have no reason to believe he cares about us, or is even aware of our existence.

Even if he is aware of us, we have no guarantee he is good or evil, whatever that means.

We know nothing about God.

We cannot know anything about God. Even if he spoke to us, we have no way of knowing he was telling the truth. Now perhaps there are vast intelligences loose in the universe, beings with immense powers and abilities, capable of manipulating matter and energy, space and time. To us, they would appear Godlike. But to themselves, they would just be more advanced beings. We are more advanced than insects, or bacteria, but we are not Gods either.

We create God in our own image, but we demand that he be omnipotent, omniscient and immortal. We do this because we are weak, ignorant, and we are dying. Get over it Johannes. You're too old to believe in Santa Claus. Face the universe like a man, not a frightened and helpless child.