Edit: I realized that my arguments have some major flaws. While true, the background knowledge required to understand the concepts involved requires that the reader be an atheist/Objectivist before he reads them. So, I appreciate the compliments, but stand by for a rewritten version designed to respect the silly theist’s hierarchy of knowledge.
(1) God is arbitrary.
There is no evidence for a supernatural being with the properties generally ascribed to God. If someone makes an assertion for which there is no evidence either way, the logical thing to do is simply dismiss it, just as I would dismiss the assertion that there is an invisible pink elephant floating over me. If this were the only argument against God, I would not be able to prove that God did not exist — but you would not be able to prove that he does exist either. Hence the claim must be thrown own as arbitrary.
Comment: The reason there I say that is no evidence for God is NOT that I cannot see him. I have never seen Australia either, or my mind, or anger, or Neptune. The reason there is no evidence for God is that the characteristics ascribed to God contradict the rest of my knowledge about the world, and cannot be integrated with it (for example, I know of no intelligence that has no physical basis, or entity that spans the whole universe)
(2) God has no identity.
Everything that exists in the universe has a particular nature, and only that nature. Things are what they are – with certain properties that have certain characteristics. Every existing thing behaves in a certain way according to its nature. (Law of causality) More fundamentally, the whole notion of something existing means having a certain nature – a particular, limited nature that is one thing and not another. Everything that exists, exists as such. In other words, non-contradiction. Something cannot be two conflicting things at the same time, in the same place, and in the same respect. God is a contradictory concept – he has no nature, no (finite) identity, and no particular causal connection to the rest of the universe. Thus, God is not only an arbitrary, but also a contradictory concept, and thus impossible. You cannot argue for a contradictory concept since the notions of proof, reason, or evidence rest on the validity of logic. (Because just like everything else in the universe, logic also has a particular nature. A is A.)
If God has a particular identity, then at most, he must be an ancient, very powerful robot playing tricks with our fate. I don’t know any theists who would argue that God is a robot. Besides, if RoboGod has to play by the rules as we do, we can beat him.
(3) God is contradictory.
Many of the traits attributed to God are self-contradictory. For example, God is omnipotent, all good, and all knowing, yet evil exists. Also: God is everywhere and nowhere. Also: God loves us and sends us to hell. Also: We have free will, yet we are pre-destined. Also: immortal soul, yet we seem to be created from scratch at birth with no memories (making re-birth pointless) Also: God defines right and wrong, yet is able to change it (same as the making a stone to big to lift thing) Also: God is actually three gods, yet he is one. Also: Man is evil, yet he was saved, yet he really isn’t saved, yet God will probably forgive us anyway. Also: God act by miracles, yet he creates physical laws, so he needs none. Also: God is concerned about our fate, yet he already knows exactly what will happen. Also: Man is in God’s image, yet he is sinful. Also: Religion is supposed to make one happy on earth, yet earthy life is about suffering and sin. Also: God wrote the sole, absolute, and unchanging source of morality, yet his Book is full of contradictions, and things you’d probably say he’d oppose today (such as stoning for adultery). Also: Pride is evil for man in heaven, yet good for man on earth. I could probably think of a few dozen more contradictions, but you get the idea. God isn’t even consistent with himself. Let’s not forget: One should have blind faith in God, yet you are about to attempt to use reason to prove me wrong.