Another traditional argument claims that there is a logical incoherence involved in certain concepts of God. This can either rely on an internal contradiction in a single attribute, or else in a contradiction in the combination of divine attributes. The first is best known in the question 'can God create a rock so heavy that He can't lift it', and the second includes problems with whether an omniscient God can make free decisions.(1) As to the argument, "Can God create a rock so heavy that He can't lift it?" I think Philippians 2:5-11 says "yes":
Have among yourselves the same attitude that is also yours in Christ Jesus, Who, though he was in the form of God, did not regard equality with God something to be grasped. Rather, he emptied himself, taking the form of a slave, coming in human likeness; and found human in appearance, he humbled himself, becoming obedient to death, even death on a cross. Because of this, God greatly exalted him and bestowed on him the name that is above every name, that at the name of Jesus every knee should bend, of those in heaven and on earth and under the earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father.I find this answer very intellectually satisfying. God the Son created the universe (John 1:3), yet in the Incarnation, emptied Himself in a radical way that we don't understand, taking on a mortal Human Nature - a body capable of suffering and dying. The God Who wants for nothing of His own nature felt hunger, pain, weariness, and the like - He became "like us in all things but sin." So in His Divine nature, He created the universe, including things His Human nature couldn't lift.
(2) As to whether and how an omniscient God can make free decisions, yes. But His free decisions aren't made within time. There are many cases where God, without violating His own Goodness, could do one of multiple options. He chooses a single one. St. Thomas Aquinas answered this question beautifully in the Summa, using Matthew 26:53 to show that God could have done what He did not.
This boggles the mind (for the same reason that predestination boggles the mind, since it seems to eliminate free will). But St. Augustine answered this in the 4th century - a God who created time and exists outside of it is bigger than these silly arguments. Note that God's description of Himself is pure Being: "I AM WHO AM." While we proclaim the glory of God the Trinity "as it was in the beginning, is now, and always shall be" in the Glory Be, from God's perspective, it's the eternal present. That's why Christ uses such a strange tense in John 8:58. Moses grasps this point well, as Psalm 90:2 reflects -- but then, he's the one God revealed it to (Exodus 3:14).
(3) A related argument:
Patrick Grim has argued that God's omnipotence and omniscience are both internally contradictory, as well as facing problems when combined with each other and further attributes. His primary argument relies on the view that certain tasks are 'essential indexicals', where the ability to complete such a task cannot be separated from self-reference. These follow from obvious and popular cases such as the rock mentioned above, and include statements like 'A snowflake falls through no effort of an omnipotent being'. This case is chosen as something that a non-omnipotent being can bring about, but not an omnipotent one.The question of whether God can cause a snowflake to fall without the effort of an omnipotent Being is really asking, "Can God cause something without God causing it?" The question is meaningless and self-contradictory. More than that, it's been answered centuries before Patrick Grim was born. Again, from St. Thomas Aquinas:
Now nothing is opposed to the idea of being except non-being. Therefore, that which implies being and non-being at the same time is repugnant to the idea of an absolutely possible thing, within the scope of the divine omnipotence. For such cannot come under the divine omnipotence, not because of any defect in the power of God, but because it has not the nature of a feasible or possible thing. Therefore, everything that does not imply a contradiction in terms, is numbered amongst those possible things, in respect of which God is called omnipotent: whereas whatever implies contradiction does not come within the scope of divine omnipotence, because it cannot have the aspect of possibility. Hence it is better to say that such things cannot be done, than that God cannot do them. Nor is this contrary to the word of the angel, saying: "No word shall be impossible with God." For whatever implies a contradiction cannot be a word, because no intellect can possibly conceive such a thing.Catholics readily affirm that omnipotence is bounded by logical coherence, calling Christ the Divine Logos. So we'll gladly affirm that God can't do x without doing x. But that's not because of any limits on God's power, but because that's a meaningless statement. Even if you hypothesized the creation of alternate worlds, one in which God does x, and one in which He doesn't, He's still doing x in one of the worlds. So atheists haven't disproved God. They've just run headlong into the Law of Identity and the principle of contradiction.
So it isn't that God is somehow less than omnipotent, but simply that the question is logically impossible. Now, Grim attempts to get around this with the self-refuting argument that man can cause snowflakes to fall without Divine assistance, so there's something we can do that God can't. There are two responses. First, he's fundamentally wrong. Given the nature of God's omnipotence, all things have, as secondary causation at the very least, God's Permission. If God does not actively Will it, He passively Permits it, which still makes the doing of the action contingent upon the Divine Being. Atheists know this - it's why they blame the sins of man on God. We have free will because God permits us to have free will. So even when we sin, we could not do so without an omnipotent Being, God, permitting us to have the freedom to do so. The very nature of God's omnipotence requires that nothing can occur which He could not stop from occurring. So for Grim to assume, a priori, that a man (we'll call him Carl) can do anything without God's assistance, is to start out assuming that God is not omnipotent, in order to prove that God is not omnipotent. If God is omnipotent, then the idea that Carl can make a snowflake fall without God's assistance is logically impossible, and Grim's argument fails. So this is a disproof of God's omnipotence only if God isn't omnipotent... which is to say, it's a lousy disproof.
Secondly, what Grim is proposing isn't even a parallel argument. He's uses something he thinks is logically possible (Carl doing something, and God not doing it) to try and show we've got a power God doesn't... but the power he's contrasting it with is the power to do the logically impossible. Can A do x without A doing x? No. Can A do x without B doing x? Perhaps (although in this case, no, as I explained in the first answer).
Those are three of the arguments which seem, on face, to be strong against God's omnipotence. None of them are, on examination. I welcome comments, rejoinders, and other vexing theological questions.