The formal solution may be presented as follows:
Let O be the set of things god can do.
1. O is a universal set
Proof: Definition of 'omnipotent' as naively stated.
2. O complement is the set of things god cannot do
Proof: Definition of the set complement - for any set S, its complement, S complement, is the set whose members are not members of S.
3. O complement is an empty set
Proof: Complement laws - the complement of a universal set is an empty set
Principle of extensionality - two sets are equal if they have the same elements
4. O complement is a subset of O
Proof: Properties of empty set - for any set X, the empty set is a subset of X.
Conclusion: The set of things god cannot do is an element of the the set of things god can do -- hence god is omnipotent.
Numinus, you have a logical error in your reasoning.
Let U be the universal set of all possible things that can be done.
Let O = set of elements of all things God can do.
Let f be an element of U = "God can create a stone that he cannot lift."
The element f represents a fallibility or failure.
Agnapostate's argument can be stated,
1. Either f is an element of O or f is not an element of O.
2. If f is an element of O then God has an element of fallibility.
3. If f is not an element of O, then O is a proper subset of U, and God can't do all things.
4. Therefore the set O either consists of a fallibility or it is not universal.
Numinus, where your logic fails is in what do you do with the element f?
In your step 3. You say O complement is empty.
Therefore f must not be in O complement, because O complement is empty.
Therefore, since O and O complement are disjoint, f must be in O.
Therefore O has an element of fallibility.
Your statement 3. is in error, because O complement is not empty.
Therefore your statement 4. O complement is a subset of O is a logical contradiction.