Are you arguing against me, or with me? Because you pretty much defined why logic cannot be used to define God and took it one step further and explained why nothing can be used to explain it... so there's no point trying to explain it and no sense living by such an unknown.
Life is full of unknowns. We all live our lives by unknowns. We all make our assumptions. Being an atheist does not mean you know everything. You do not have a choice in this. The question of whether God exists is the question of the nature of our own selves. We know we are conscious decision makers, or at the very least that we are aware and perceive some sense of being conscious decision makers. All of religion evolves from there. I don't think I overstate the case in saying all of philosophy and ethics also starts from there.
One of the most fundamental things about being human is utterly invisible, and yet irrefutably exists -- our own conscious 'mind'. Whether it is a product of the physical world, or the physical world is the product of a mind, or some bizarre as yet unrecognized combination of the two, no one can say. But you can't just say that you don't believe in anything that cannot be fully explained, because you really have no choice. There are things in life that are, yet are not fully explained.
By the time you start combining Goedel's proof with chaos and fractals, you get to a point where it seems very likely that there is a theoretical limit to knowledge. We are closer to proving that it is impossible to know everything than we are to explaining everything.
This is a world where faith must always play its role.