i) Can an omnipotent being make a stone so heavy that he can't lift it? The stone paradox takes the form of a dilemma:
Either God can make a stone that he can't lift, or he can't make a stone that he can't lift.
If he can make a stone that he can't lift, then he's not omnipotent (inasmuch as he can't lift the stone).
If he can't make a stone that he can't lift, then he's not omnipotent (inasmuch as he can't make it).
Either way, there's something he can't do. So whether you answer yes or no, God is not omnipotent.
ii) Of course, this may be a false dilemma. If the proposed task is really a pseudotask, then inability to perform a pseudotask is not a limitation on omnipotence. Rather, it's disguised sophistry.
iii) Sometimes the debate turns on the definition of omnipotence. There are many things I can do that God can't. For instance, I can sneeze, but God can't. Is that a problem for omnipotence–or for a silly definition of omnipotence?
iv) Apropos (iii), if God chooses to work through natural means, then that's a self-imposed limitation on what he can do. Natural means are finite. Natural means have inherent limitations. If he chooses to work through natural means, then his chosen medium limits what he can do.
God can work apart from means. He's not limited to means.
v) As formulated, the paradox has anthropomorphic connotations. God can't lift a stone the way a man can lift a stone. God doesn't lift a stone through muscular exertion, or pulleys, or a crane. Perhaps we should substitute a less anthropomorphic verb like "levitate."
vi) Lifting is a twofold relation:
a) Moving an object from one place to another
b) Moving an object from a lower to a higher position
vii) If the stone is the size of the universe, then it can't move, for it occupies all the available space. It can't shift from one location to another if it takes up all the free space. There's nowhere for it to go. A stone the size of the physical universe is immovable. It can't change position. There's no room for the stone to be displaced.
viii) Likewise, lifting something presumes a frame of reference. To lift something is to pick it up.
An agent can't lift a stone in outer space, because the stone has no frame of reference in relation to which it's higher or lower.
Likewise, does it make sense to say an agent can lift a stone off the ground that's bigger than planet earth? In what sense is it higher than a round object if it's the same diameter (or greater) than the round object? Given the curvature of the reference frame, what makes the stone higher rather than sideways or underneath?
It only makes sense to say it's higher if the earth is flat or the object is so small in relation to the globe that the point of contact is virtually horizontal in relation to the vertical action (raising the stone to a higher position vis-a-vis the ground).
ix) I assume the strength of the atomic bond (chemical bonding) naturally limits on how big a physical object can be. I assume there's an upper limit on how large a stone can be. Beyond a certain threshold, the attractive force is too weak to keep the atoms and molecules from shearing.
Although God could make a miraculously large stone, that interjects equivocation into the paradox. It's not a real stone. By "stone," we usually mean a natural object.