The Stone is a Paradox?

For those who are unfamiliar with the Stone paradox, it is basically this: Can an omnipotent being create a stone so heavy that said being cannot lift it? 

It's essentially an argument against omnipotence being tenable since the question posed seems to give a contradictory response either way you answer the question.  If an omnipotent being can create a stone so heavy that said being cannot lift it, then said being is not omnipotent.  If an omnipotent being cannot create a stone so heavy, then said being is not omnipotent.  Thus we have the paradox.  There are other arguments against omnipotence, but I want to focus on this one. 

Introductions aside, there is something I felt wasn't quite right with the argument… not that feelings really have any say about it.  My question concerning the paradox is whether the question posed is a valid question, i.e. is it loaded or the likes?  So I tried to think of an appropriate analogy.  Since omnipotence deals with the infinite, I decided upon using the set of rational numbers (here forward known as Q). 

Here's the analogous question I pose: Can the set Q contain a rational number so large that said number cannot exist in Q? 

Any mathematics degree holders will tell you the answer is no and prove it so.  Would we then assume that the set Q doesn't really have the quality we assume, or alternatively would we assume that such a hypothetical number does not exist and therefore the answer is no by a vacuous argument?  Now, I admit there are some fundamental differences in the analogy given.  But let's use the ideas that might have been triggered by my analogy to try to understand the Stone paradox a little better. 

The Stone paradox starts out with the assumption that an omnipotent being (here forward referred to as OB) exists.  We can then consider that all things within the power of OB could do, namely everything – hence the transliteration 'all-powerful'.  However, there is one constraint that we would put on such power.  That is that an OB is restricted to logic.  What I mean by this is that OB cannot do that which is logically impossible, such as an OB cannot create a square circle (or squircle).  A square circle has no meaning and is logically absurd.  What is a square circle anyways? 

Now, within the set of things that an OB could do is certainly create a stone of some finite weight, and should certainly be able to lift that stone.  Just as certainly a rational number exists in the set Q. No matter how big a stone could be created, it will always have a finite weight* and so an OB should always be able to lift it by the simple principle that whatever finite weight (or force) that the stone has, we can assign a force to OB greater than the weight by a factor of, oh, say 10 (arbitrary number greater than one) which is well within the limits of infinity and would then automatically be sufficient to lift the stone.  You can continue to increase the stone's weight ad infinitum and you could still conceive a greater force than the weight of the stone.  So if the question were to be "Can an omnipotent being create a stone of any weight and then lift it?", we would have to answer yes. 

Return, then, to the original question: Can an omnipotent being create a stone so heavy that said being cannot lift it?  What is it we're really asking here?  Are we asking about the stone or are we asking about whether an OB could lift it?  Certainly the stone is mildly arbitrary and as we already saw an OB can create a stone of any weight and then lift it.  We then seem to be asking whether an OB can do something such that the OB cannot do something.  Is that a valid question or not vacuous?  I don't think so, and I'll do my best to explain. 

No matter what size stone an OB can create, it will always have a finite weight*, as discussed, and so the OB will always be able to lift it.  So we ask to create a bigger stone, and a bigger one, and so forth till we get to a stone that the OB cannot lift.  But this is absurd.  As discussed, the OB can always have more force than a conceived stone's weight just as no matter how big a number you can think of, there's at least one bigger (actually there's an infinite number bigger).  So it's no longer a limitation of the OB's power, but a limitation of the weight of a stone.  No such stone could exist such that it would have more weight than what we could conceive a force having, just as no such rational number can exist that it is so large that it exists outside Q.  So by a vacuous argument, since no such stone could exist, the question is logically invalid because it requires a stone to have an 'infinite' weight– what we posed was necessary to not be the case when we consider an OB's power to have the restriction of logic. 

But perhaps there's something else that an OB can do so that the OB cannot do something.  We could ask: "Can an OB create a number so large that the OB cannot count in numerical order to it given as much time as needed?"  But again, such a number would have to be finite, and so the real question would again be "Can an OB do something such that an OB cannot do something?"  But doesn't this seem to be a loaded question?  You're both assuming that the OB is both all-powerful and not all-powerful, so of course you're going to get a contradiction.  It'd be similar in asking how nothing can have the property that it has no properties – it's a kind of word game.  What I pose is this: In the set of things an OB can do, there is nothing that would automatically lead to an OB not being able to do something else.  Perhaps I'm wrong, and I'd like to see an example of something that would make this truly a paradox. 

*A stone with an infinite weight has no meaning.  Infinite is not a number and so cannot  be assigned to a measurement for anything except perhaps abstractions, which even then tends to be non applicable as a measurement especially when you consider that there are 'different sizes' of infinity.