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This is something I wrote about a decade ago, on my old website which no longer exists. I still like it, so I reproduce it here with some minor edits.

What I am about to write should not really be necessary. I am certain that a satisfactory discussion of the subject exists elsewhere, and the matter is simple enough that anyone who feels compelled to find the answer can do so on his own, given some time. However, the frequency with which this question comes up disturbs me, and I am a little weary of repeating to myself the same line of thought every time I see it. Besides, I like my own answer better than someone else's. So I will write this, and the next time I see the question, I will simply follow the link in my mind to this essay.

It is often asked, "Can God create a rock so heavy that he cannot lift it?" There are variations, of course, such as "Can God create a being more powerful than himself?", "Can God decree that he will be no longer omnipotent?", etc. But they are only variations, and my analysis of the first question easily translates into the others. Presently, therefore, I will concern myself with only the first question.

Following the question, it is asserted that the question can be only answered yes or no. If the answer is no, then God cannot create such a rock, therefore he is not omnipotent. If the answer is yes, then God cannot lift such a rock, so again, he is not omnipotent. It is then impossible for God to be omnipotent, but the Christian God is said to possess this attribute. Therefore, the Christian God cannot exist.

Let us now see how a Christian can respond to this argument.

First, the precise meaning of "omnipotence" must be determined. In particular, it should be determined whether or not omnipotence entails being able to defy logic. If it does, God's omnipotence can be preserved by asserting that God can both create a rock that he cannot lift, and lift it. This would imply that logic is a creation of God: God created it to govern the universe, but it doesn't apply to him, and he overrides it at will. The Christian can simply choose this definition of omnipotence, and the debate of whether the Christian God can exist is technically over. The nonchristian may assert that this is the incorrect definition, but the Christian has the prerogative to decide precisely what "omnipotence" means in his own belief system. However, the Christian is now forced into the view that logic is necessarily a creation of God (which may or may not be true. All else being equal, it is better to leave this issue unresolved, for the sake of flexibility). In order to avoid this situation, it is desirable to demonstrate that God can maintain his omnipotence even if omnipotence does not entail defying logic.

If defying logic is not a part of omnipotence, the argument of the previous paragraph doesn't hold. God could not both create a rock he cannot lift, and lift it. That would be a logical impossibility. In this scenario, logic is an attribute of God, rather than his creation. God would be "he who is logical", and he could not be illogical, since that would mean God is not God. At this point, it may be asked "Can God be illogical?" or "Can God be not God?", in a further attempt to show that if God is to be bound by logic, then he cannot preserve his omnipotence. But these questions are just simpler versions of the original question that we are trying to answer. Rather than waiting until the end of the analysis to answer these variations, We will address them now, where they can serve to illuminate how the original question about the rock can be answered.

The more elemental question of the two is "Can God be not God?" The answer is, "No. That would be a logical impossibility". But does that not mean God is not omnipotent, since he cannot be not God? No, because under our current definition, omnipotence does not include the ability to do the logically impossible. And clearly, it is a logical impossibility and meaningless nonsense for anything to be both God and not God. So the question, "Can God be not God?" cannot discredit the Christian God's omnipotence. Likewise, any question in the same form is equally ineffective. Now it remains to be shown that all other questions that were previously mentioned can be reduced to this form.

Consider the question, "Can God be illogical?" Among the properties that God has, the relevant one in this case is "he who is logical". Substituting this meaning for the word "God", we get: "Can he who is logical be illogical?" So the true construction of the question is seen, with its clearly self-contradictory nature.

The original question about the rock that God cannot lift is only slightly more complicated. Let us start with the exact question: "Can God create a rock so heavy that he cannot lift it?" Here, "he" is referring to God, who can lift any rock. Substituting this description of God for the word "he", we get: "Can God create a rock so heavy that he who can lift any rock cannot lift it?"

"A rock so heavy that he who can lift any rock cannot lift it" is the logically impossible element in this question. God cannot create such a rock. It would be a logical impossibility. God is still omnipotent, since omnipotence does not include the ability to perform logical impossibilities.

So, the omnipotent Christian God can exist, with either definition of omnipotence. It remains to be determined which definition is the correct one. My personal belief is that logic is an attribute of God, and therefore omnipotence does not include the power to break logic. But that is another matter for another time. For now, the question at hand is settled.

You may next want to read:
A real discussion on the problem of evil and omnipotence.
Another post, from the table of contents