Specifically, I think the idea of classifying all properties as "good" or "not good" is hopeless: I do not believe there is any such classification that fits anyone's intuitive ideas about goodness well enough that calling something "godlike" if it has all "good" properties is credible. More specifically, I suspect it's consistent to suppose -- hence there are no humanly-comprehensible counterexamples -- that the only notions of goodness obeying all Goedel's axioms are ones that look like "P is good iff P(x)" for some fixed object x. (And, like Krishnaswami, I think the system of modal logic Goedel needs is awfully strong.)
In the spirit of Anselm's ontological argument, however, I offer the following proof of the nonexistence of God:
Consider a really bad argument for the existence of God. In fact, consider one so bad that no worse argument can be conceived.
Obviously a bad argument for something fails to prove what it purports to prove. But merely failing is a pretty mediocre kind of badness. The worst possible argument, surely, has to be much worse than that: it must conclusively prove the opposite of what it's meant to prove.
Now, the worst conceivable argument for theism clearly "exists in the understanding", as St Anselm put it. But it can't exist only there -- because a bad argument is more damaging to the premise it's meant to support if it's actually made.
Therefore, there is an argument for the existence of God which is actually a conclusive proof of the nonexistence of God.
And, of course, any proposition that can be conclusively disproved is false; therefore there is no God.
(This argument is in my opinion almost exactly as strong as the original ontological argument for the existence of God. Which is to say, it's absolutely hopeless. But I think it's fun.)