The unfortunate thing about math is that pathological and counter intuitive stuff is actually the most common type of stuff in our mathematics, we just have a hard time finding them because we're wired to think of objects with nice properties. Convergence is just another one of the many casualties of this war that pathological objects and paradoxes like to wage on our weak human minds. First they took away our rational numbers, which pissed Pythagoras off. Then they took away our countable sets, with Cantor's diagonal argument, which was fine for a while, except now we have Skolem's paradox, which asserts that all sets from ZF set theory ARE countable. Oh, by the way, the word "paradox" is a misnormer; all sets *really are* countable, including the reals, and this in no way contradicts the fact that our model of set theory has the uncountability of the reals as a theorem. And if you reject the axiom of choice, the real numbers is actually a countable union of countable sets. We also have weird stuff like the Weirstrauss function which is continuous everywhere but differentiable nowhere. And it turns out, "most" continuous functions aren't differentiable at all whatsoever. Our brains just aren't wired to find such pathological functions. And don't even get me started on the Cantor set, which has measure zero but is somehow uncountable. And, on the topics of measure theory and set theory, we have the whole Axiom of Choice dilemma. If we accept it, we get the Banach-Tarski paradox where you can basically chop up a pea and reassemble it so that it's bigger than the entire milky way if you wanted to. But if we reject it, we lose soooooooooooo many nice things, like every vector space having a basis (and every basis having the same cardinality. How do you even define the dimension of a vector space without those two results?). We'd also lose the Hahn-Banach theorem. These last two things would impact hilbert spaces too, so any discipline that uses Hilbert Spaces would be affected (actually, I'm not too sure if physicists would care or even how much this would affect them, but still...). The axiom of choice, by the way, implies that the real numbers are well ordered. In other words, there's some ordering in which every bounded subset has a minimum and maximum, not just an infimum and supremum. With our standard, intuitive idea of ordering numbers (i.e. the way we decide which numbers are bigger than other numbers) this is impossible, so this means our idea of what it means for one number to be bigger than another is inferior in a sense. There's a better way to order than the one we use. What does that say about humans when the way we decide if one number is bigger than the other isn't actually the best way to do it? That's such an intuitive idea that we're all born with, and we may actually be wrong about that. Which reminds me, depending on how arithmetic are formulated, there may be an infinite amount of "non standard models". It's kind of like how, for example, there are 5 groups of order 8. Well, there are multiple systems of natural numbers (an infinite amount, to be exact) if you try to formulate it in terms of first order logic (the same kind of logic that tells us that the reals are countable). So not only are there an infinite number of models of numbers other than the one we use, but our theory of numbers can't even prove its own consistency because Godel hated everyone and wanted to explode our brains. Speaking of which, there's Godel's theorem, where our whole idea of "truth" may be shaken. In some systems, if they are consistent, they must have undecidable statements. Philosophers argue about whether that's a big deal in the grand scheme of things, but if this does affect epistemology in any way, it's going to be in a negative way for sure. The fact is, math has been waging war on our intuition for as long as we've been able to think. I say it's time we fight back! I'm not sure how we wage war on abstract concepts and theories, but we should at least try! lol