We're working on Patrick Grim's "Impossibility Arguments" in The Cambridge Companion to Atheism in my Atheism course. Here's a particularly striking argument:
Because the [impossibility] arguments at issue operate in terms of a set of more or less clear specifications, of course, it is always possible for a defender of theism to deflect the argument by claiming that the God shown impossible is not his God. If he ends up defending a God that is perhaps knowledgeable but not omniscient he may escape some arguments, but at the cost of a peculiarly ignorant God. The same would hold for a God that is perhaps powerful but is conceded to be less than omnipotent, or historically impotent but not literally a creator. If the term "God" is treated as infinitely re-definable, of course, no set of impossibility arguments will force the theist to give up a claim that "God" in some sense exists. The impossibility arguments may nonetheless succeed in their main thrust in that the "God" so saved may look increasingly less worthy of the honorific title.
A more frequent reaction, perhaps, is not redefinjtion but refuge in vagueness: continued use of a term "God" that is allowed to wander without clear specification. Here as elsewhere - in cases of pseudoscience, for example - resort to vagueness succeeds in deflecting criticism only at the cost of diluting content. If a believer's notion of God entails anything like traditional attributes of omniscience, omnipotence, and moral perfection, the force of impossibility arguments is that there can be no such being. If a believer's notion of God remains so vague as to escape all impossibility arguments, it can be argued, it cannot be clear to even him what he believes - or whether what he takes for pious belief has any content at all.
The whole article, with several arguments for why omniscience and omnipotence are impossible, is here:
Patrick Grim, Impossibility Arguments
Grim surveys several of the most recent, most logically sophisticated accounts of omnipotence and omniscience from Flint and Freddoso, Rosenkrantz and Hoffman, and Wierenga. None of the explanations work, he argues, because they either fail to be of sufficient scope to be worthy of God, or by being overly ambitious, they collapse under logical counter examples. That is, God's properties, whatever they are, must be sufficiently maximal. God, in order to be God, must have as much knowledge and power as can be had. But on the best accounts we have, omnipotence and omniscience are anemic and mundane beings could qualify. The most knowledge and power that any being can have are not enough to be God worthy. The result, suggests Grim, is that after thousands of years of grappling with the problem, we still don't have a clear account of what it would be to be omnipotent or omniscient. The implication is that we should conclude that the properties are impossible, unless the theist can produce some account that makes sense and that clarifies his claim that he believes in such a being.