I suppose I am assuming that the universe operates under some set of formal rules (though they might not be deterministic) independently of our ability to describe the universe using formal rules. I would also say that our inability to comprehend a given contradiction is related to the fact that we are inside the system. If God were outside the system he would not necessarily have this problem.
I disagree with your second point, though. Sure, 1 and 2 are labels for concepts that exist within a formal system we've developed, and sure, we can create an isomorphism to different labels. But I would consider this to be the same formal system. The example I gave (working in the integers mod 2) involves switching to a formal structure that is decidedly not isomorphic to the integers under addition.
Also, sorry if I was unclear - I did not mean to imply that mathematical formalisms as we've developed them are related to the fundamental laws of the universe. I only meant to say that if the universe is a formal system of some sort, and God operates outside that formal system, then it is conceivable that God could switch to a different formal system where things that we consider impossible are not, just like we can switch to a different formal system where 0 and 2. Maybe God could do something analogous and put me in the universe (mod 10 feet) so that if I walk ten feet straight across the room I'll end up where I started; this seems like a contradiction in our universe but is definitely imaginable.
[Quick edit for clarity: maybe it doesn't seem like a contradiction that I could walk ten feet away and end up back where I started, but it does seem like a contradiction that I could walk ten feet and both be ten feet away, and also be exactly where I started. This is what I imagine happening in the universe (mod 10 feet).]
The universe with the 10-feet torus topology would certainly be a different universe governed by different laws. Still, one could conceive of a formal system of addition which would be exactly same as our present one, only it would not apply to distances (in a straightforward way). The same way as we can conceive the addition mod 2 arithmetics.
As for the seeming contradiction, if you define "p being x feet away from q" as "there is a geodetic of length x connecting p and q", then obviously "I am ~40,000 km far from Istanbul while I...
[This is a draft intended to be developed into a top-level post - it wouldn't feel wrong to make it such right now, but it wouldn't quite feel right. I am not entirely sure how to end it or if I could generalize better at the end. I kind of like the ending I have, but I'm not sure if the point overall is coherent enough. Thoughts/suggestions/criticism would all be appreciated. ETA: The problem here may be that this is actually a follow up (or a footnote) to another article I've been thinking of about Weasel Words and the art of misleading through langauge; related to my earlier post on Not Technically Lying]
When I was a teenager, I remember hearing a couple of riddles that I thought were neat:
"Could God draw a square circle?"
"Could God create a stone so large that even He could not lift it?"
Let me just disclaim that this post has pretty nothing to do with religion. I just think that these are great examples that many people may be familiar with. That said, consider: do either of these problems pose a threat to the existence of an omnipotent God?
The answer, as will be clear on a full exposition, is a resounding "No." These are terrible, awful, misleading arguments, and the second one illustrates a relatively common trick used to sneak past an audience's intellectual defenses.
These riddles both fail to provide relevant counterexamples for the exact same reason, even though the second may seem to make more sense. The first is simpler: a square circle is not a thing. In a practical sense, we can put the words next to each other, but there is simply no way to translate the sound "square circle" into some kind of expectation or thing in the real world, in the same way one could translate, "red barn" or "white unicorn" into an expected observation. It is impossible for anything to be both square and circular, so the fact that God cannot do something that cannot be done does not limit His omnipotence. By the same token, God could not create a married bachelor (using the strict definitions of the terms), as a bachelor is an unmarried man. The inability to violate the law of non-contradiction does not appear to be a legitimate refutation of omnipotence. If we taboo, "square circle," there isn't really a meaningful way of describing the thing you are insisting God be able to draw.
"A stone so large that God cannot lift it," is exactly the same thing as a square circle. It sounds like a problem, since it's showing that God can't create a big enough stone. But an omnipotent being could presumably lift an object of any arbitrary size. Therefore, no stone could ever meet these criteria. If we taboo "so large that God cannot lift it," there is no actual weight you could describe such a stone as having. Presumably, God could lift a stone that weighted 3^^^3 tons, or even 3^^^^^^3 tons. You've created a hollow adjective: a descriptor whose actual meaning makes an argument self-evidently bad, but which is appealing if you don't actually think about it. It's not Not Technically Lying, because it isn't untrue, it's meaningless, which makes it harder to detect (though less common).
This is an extreme example. Usually, hollowness allows a speaker to be vague enough that they sound like they have a point when a clear definition of their terms would disprove this. Offenses in common language are usually a bit less egregious. "The president hasn't done enough to fix the economy," comes to mind as an example. What exactly, should he have done? There has probably never been a president in history whom people would generally agree has done "enough to fix the economy;" indeed, most economists would question the power of the president to seriously influence such things. "Hasn't the president failed to end the recession?" may be technically true, but it isn't really useful to call someone a failure for not doing something they lack the power to do. This example is merely illustrative; it is often easy to create descriptors that make your conclusion apparently foregone, despite their actual lack of substance.
Using such slanted terms is among the darker of the Dark Arts. It plays on its audience not by appealing to the irrational vagaries of the human mind; such efforts are, at least, often transparent. Rather, it masquerades as a rational argument, requiring complex nuance to refute. For those who are not disposed to disagree, it can escape the defense mechanisms of even a cautious mind. Understanding this concept can make it far easier to pinpoint the error in some beguiling arguments.