God vs. Logic

Tonight, after four CS103 (Logic) lectures, I tried to disprove the existence of God. My strategy was to use one of the many paradoxical facets (e.g. Free Will) of Christian theology to show that the contradictions precluded the existence of a supreme being. It's probably already been pointed out to you that if God is omnipotent He could create something He was incapable of destroying or lifting, which would make him im(omni)potent. One way to argue this paradox is by applying a time dimension, i.e. God can create something He cannot destroy, but as soon as He creates it, He is capable of destroying it. Or some quasi-state in which God is both capable and incapable of destroying it. But that's not very logical. Plus, the whole point of bringing up these conundrums is to watch the religious people squirm. So I thought, why not throw insult to injury? Maybe if I could couch this inconsistency in a formal logical proof, replete with valid inferential justifications and esoteric logic laws (e.g. Modus Ponens), I could further fluster the novice theologians of the world.
I make no claims as to the novelty of this enterprise (I'm sure there's a complete logical dis-proof of God out there somewhere, but I will refrain from searching for it for the time being, to ensure my work is 100% original. So, without further ado, I present my first attempt to ascend the Tower of Babel:

I begin with five premises, which I hope are incontrovertible.
Note:
P(x) → Q(x): If P, then Q.
∀x P(x): P(x) for every x
∃x P(x): There exists an x such that P(x)
¬ P(x): Not P(x)


1. If God exists, He is omnipotent.
Formally: If x is God then x is omnipotent.
G(x) → O(x)

2. Only God is omnipotent.
If and only if x is God is x omnipotent.
G(x) ↔ O(x)

3. An omnipotent being can create anything.
If x is omnipotent, then for anything y, x can make y.
O(x) → ∀y C(x,y)

4. An omnipotent being can destroy anything.
If x is omnipotent, then for anything y, x can destroy y.
O(x) → ∀y D(x,y)

5. An indestructible object cannot be destroyed.
An object x that is indestructible cannot be destroyed by anything y.
I(x) → ∀y ¬D(y,x)

Now, from these five premises, I infer the rest...

6. If an indestructible object exists, then nothing can destroy it.
If there is an indestructible object x, then there is an object x which no thing y can destroy.
∃x I(x) → ∃x∀y ¬D(y,x) [Existential Generalization, 5]

7. If an indestructible object exists, then there is an object that all things cannot destroy.
If there is an indestructible object x, then for any entity, there is an object it cannot destroy.
∃x∀y ¬D(y,x) → ∀x∃y ¬D(x,y) [Magic x for all y, 6]

8. If there is an object that all things cannot destroy, then no thing can destroy all objects.
∀x∃y ¬D(x,y) ≡ ¬∃x∀y D(x,y) [De Morgan's Law for Quantifiers, 7]

9. If an indestructible object exists, then there is no thing that can destroy all objects.
∃x I(x) → ¬∃x∀y D(x,y) [Hypothetical Syllogism, 6,7,8]

10. If an omnipotent being exists, then a thing that can destroy anything exists.
∃x O(x) → ∃x∀y D(x,y) [Existential Generalism, 4]

11. If there is no thing that can destroy all objects, then an omnipotent being does not exist.
¬∃x∀y D(x,y) → ¬∃x O(x) [Modus tollens, 9, 10]

12. If an indestructible object exists, then an omnipotent being does not exist.
∃x I(x) → ¬∃x O(x) [Hypothetical Syllogism, 9, 11]

13. If an omnipotent being does not exist, God does not exist.
¬∃x O(x) → ¬∃x G(x) [Modus tollens, Existential Generalism, 1, 12]

14. If an indestructible object exists, God does not exist.
∃x I(x) → ¬∃x G(x) [Hypothetical Syllogism, 12, 13]


At this point, I could continue to show that God can create an indestructible object. But then I'm stuck, because nothing stipulates that the indestructible object actually exists. And I can't include its existence as a premise, because most likely it doesn't exist.Hence, I need to find a way either to show that (a) if an indestructible object does not exist, God does not exist, or (b) convince God to make an indestructible object. Because, as I have it now (and this is the funny part), until God actually makes the indestructible object, I cannot disprove his existence!

So that's it. God can make an object he cannot destroy. He just doesn't want to. Paradox solved.