There are, however, some things that God cannot do. God can’t do anything that would deny His own character. This is why Theists describe God’s omnipotence as God’s ability to do “all His holy will.”
A modified version of Euthyphros dilemma says, "Is what is morally good commanded by God because it is morally good, or is it morally good because it is commanded by God?" As Christians we would answer neither. The moral, objective “good” is grounded in God’s nature. The perfect good is God. To move in the opposite direction is to sin, moving from the perfect "good" to “evil”. God is perfectly holy. Therefore God can’t sin. God is self evident. Therefore God can’t deny Himself. God is necessarily existent. Therefore God can’t cease to be God. God can’t act in ways that are inconsistent with His attributes.
Because God is truth, God can’t lie (Titus 1:2 – literally “the unlying God”). He can’t deceive or tempt. “Ha!” says the atheist, “I can lie; therefore I can do something that God can’t.” It becomes how you frame the question. Lying is a moral failure. So can we fail? Yes, but God is incapable of failing morally. Lying then is not an accomplishment.
Other groundings in God’s nature include logic and reason. God can’t be illogical or unreasonable. The atheist asks, “Can God make a rock so big that He can’t lift it?” This is just silly. You are trying to pit the power of God against... the power of God. This is like asking, “Who would win in an arm wrestling contest between God and Himself?” The atheist asks, “Why can’t God make a square circle?” The answer is because it is nonsense. It is like a married bachelor. It is logically incoherent.
The atheist says,
“…if this being (God) is all knowing and all powerful then it can a) see all of time, and b) do anything it wants to do. However, if it can see all of time then it can see what it will do next, which means that it is not all powerful. If it is all powerful then it can change its mind which means that it's not all knowing. As it is very easy to show, with basic logic, that a being that is simultaneously all powerful AND all knowing is an impossibility, it can be equally easily deduced that god cannot be perfect.”
This is similar to Richard Dawkins argument in "The God Delusion" that it is logically impossible for God to be both all knowing and all powerful. First let me deal with the aspect of time. It would seem that if God is the creator of time, then He is not subject to time. Although God interacts with those that are “in time” He is outside of time because He is eternal. To say that God’s next move is set and He is powerless to do anything about it assumes that God is suddenly subject to the restraints of time that He created.
The next problem is that Dawkins and other atheists who believe this to be a powerful argument for God’s non-perfection and therefore non-existence have an incorrect, and in all honesty, a pretty juvenile view of omnipotence. Now to be fair, this is often propagated by Christians who haven’t looked very deeply at this issue. The Sunday school view of God’s power is often “God can do anything He wants” without any qualifications. A better definition might be that
God is capable of doing anything that power is capable of doing. God is not limited by any lack of power to do anything that power might accomplish. - Greg Koukl
The other problem is that the atheist believes that the ability to “change your mind” is a positive ability. Again it is how the question is framed. Perfection is not a limitation. In the previous example about lying, the ability to sin is presented as a positive. Just the opposite is true. Once more, the atheist wants to present perfection as a negative; a limitation and then accuse God of not being perfect. God is omniscient. He has perfect knowledge. Here the atheist uses the ability to change one’s mind as a benefit when in fact it is the consequence of imperfect knowledge.
So the atheists’ logical process goes something like this.
You say God is perfect, right? Ok then, can God be imperfect? No? Then God isn’t perfect.