If a god is all-powerful then it follows that he cannot make another god who is more powerful than he already is. If he did then that must mean that he was not ALL-powerful after all, because by definition, an ALL-powerful god cannot create anything more powerful than himself. The paradox indicates that there can be no such thing as an all-powerful entity, in the same way that there cannot be a largest number. You can always make a larger number. Even infinite numbers can be infinitely smaller than other infinite numbers. There is no end to the infinite for even the infinite can be dwarfed.
Using logic we have shown that god is not all-powerful. At this point religious people throw their arms up in horror. They cry, How can you use logic to say anything about god? But when religious people insist that the Universe could not exist without a god they themselves try to use logic to describe their god - they claim he must the Creator of the Universe. (This is ironic, especially as such logic is deeply flawed: they cannot explain how the god himself came to be, which rather misses the point).
But supposing a god exists then could he have invented logic?
For the sake of this argument I will assume that meaningful existence can only occur embedded within a cloud of logic. Imagine a Universe in which there is no logic. There would be no meaningful numbers, you would not be able to add, subtract or even count. (At least the average graduate of the modern schools would find themselves at home there). There would be no laws, no physics, no predictability, no cause and effect. Such a world would be utterly devoid of meaning. We can assume that for any Universe in which anything meaningful can happen, logic is a key ingredient of the mix.
Likewise I will assume that you cannot invent anything if logic doesn't exist. (I admit that Microsoft Windows seems like an obvious exception to this rule, but actually there is logic behind this horrible operating system: it is a weapon to bring Microsoft world domination and $$$$). As we have seen, without logic, there can be no cause and effect. There can be no invention.
A fallacy beloved of religious people is that "god" invented logic. This crazy argument implies that there was a "time" when this god "existed" in a "place" sans logic. If there was no logic then god cannot be said logically to have existed at all, as the state of existence is a logical one. If god could not be said to have existed then it is logically a fallacy to state that he did exist before logic existed. Therefore god could not have invented logic.
Logic isn't part of the fabric of the universe in the way that matter, time and space are; logic is neither energy nor any physical thing: logic is informational. Any creator god must be an entity that processes information. Therefore he cannot have created information itself without using information to do it. So what came first, the creator or information? It is a chicken and egg scenario, not unlike the paradox of how existence can come from non-existence in the first place.
You cannot have intelligence without information and you cannot have information without logic, they are aspects of the same thing. Gods are usually deemed to be intelligent and intelligence is about processing information meaningfully. Ergo such a god could not have existed at all without information and logic existing simultaneously. Hypothetically there is no reason why information and logic cannot exist without a god but on the other hand it is impossible for a god to exist in the absence of logic and information. That is another constraint on why a god cannot be all-powerful. He cannot destroy all information without destroying himself.
After all, a god is a processor of information, just like all life, computers and, in fact, everything in the Universe (yes, even women drivers do it sometimes). To be a creator you have to have an idea, otherwise the creator has no mind or freewill: such a creator is merely a mindless slave to the laws of physics, the very laws that he was supposed to have created. (I would argue that we humans are exactly like that and that do not have freewill but that is another story covered elsewhere).
Many believers intuitively think of god as a hyper-intelligence who is observing them; pulling strings; making things happen in the background; responding to prayers; rewarding saints and punishing sinners and such like. That belief requires that intelligence is at work. As we have seen, intelligence is something that processes information. If god did not process information, then he could not have intelligence and he could not have knowingly invented anything. Therefore a god could not have deliberately invented information. Conversely, if a god had always been intelligent then he had always processed information. Therefore he could not have knowingly invented information in the first place because he could never have functioned intelligently without it. As information is based on logic, it follows that an intelligent god could not have invented logic. If a god exists then he was created by logic, not the other way around.
You might conclude that God IS information. But then who or what created information and how?
No invocation of a god can solve the riddle of existence because no intelligent god can invent logic and information from scratch. It is back to square one. Given the absence of evidence in favour of the existence of a god, it is easier to purge god(s) from the picture. God is innocent of the crime of creating our Universe until found guilty. Thus we can skip the pointless burden of having to account for the existence of god(s) and concentrate on the Universe and logic itself. At least we know they exist in the first place. Don't they?!?