The Incoherence of Omnipotence

"If God is all-powerful then God can do anything. But if God can do anything then God can create a stone too heavy for God to lift. But in that case God's not all-powerful. In fact, the very concept of the almighty is self-contradictory, and God cannot exist."

Objection 1: God's not perverse

God would never choose to create an unliftable stone, or to do anything that allowed for there to be contradiction. Reply

Objection 2: The atheist's complaint is silly, it's beside the point

When devout believers say that God is all-powerful, they don't really mean to say that God can create immovable stones. There are certain things that God can't do -- for example, create a round square -- and acknowledging as much does not detract from God's glory. Reply 

Objection 3: Mysterianism

It's true that the concept of God is self-contradictory. But so what? God is a mystery who transcends understanding, and for whom laws of logic do not apply. Replies

Summary

The Omnipotence Argument does not pretend to refute all varieties of theism. It aims to compellingly refute absolutist theism, and it claims incidentally to nullify the Cosmological and Axiological Arguments.

References

<Peter Suber>

There are also arguments against the logical coherence of God's omniscience. For a debate on this matter, see <Patrick Grim & Alvin Plantinga>.