Is God Really Intelligent?

If God is really the transcendent first being that Christians say He is, then it's silly to assume ahead of time that we can have a very thorough understanding of what He's like. We need more than just the word intelligent plus a bunch of negations, not receiving ideas, not reasoning from one judgment to another, and so forth. So we have to say and mean things like this: God is aware of himself. He's aware of the world, and so on. Even if it's not like ours, is that as far as we can go?